{"id":268,"date":"2025-04-09T17:37:55","date_gmt":"2025-04-09T17:37:55","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/5-5-zeros-of-polynomial-functions-college-algebra-2e-openstax\/"},"modified":"2025-08-21T18:56:54","modified_gmt":"2025-08-21T18:56:54","slug":"5-5-zeros-of-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/5-5-zeros-of-polynomial-functions\/","title":{"raw":"5.5 Zeros of Polynomial Functions","rendered":"5.5 Zeros of Polynomial Functions"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_1bb57d2c-789f-4252-bfe4-edc0873f1e29\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Evaluate a polynomial using the Remainder Theorem.<\/li>\r\n \t<li>Use the Factor Theorem to solve a polynomial equation.<\/li>\r\n \t<li>Use the Rational Zero Theorem to find rational zeros.<\/li>\r\n \t<li>Find zeros of a polynomial function.<\/li>\r\n \t<li>Use the Linear Factorization Theorem to find polynomials with given zeros.<\/li>\r\n \t<li>Use Descartes\u2019 Rule of Signs.<\/li>\r\n \t<li>Solve real-world applications of polynomial equations<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n[caption id=\"attachment_1551\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1551\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-1-Quinceanera-sheet-cake-300x210.jpg\" alt=\"\" width=\"300\" height=\"210\" \/> Figure 1[\/caption]\r\n<p id=\"fs-id1165137758829\">A new bakery offers decorated, multi-tiered cakes for display and cutting at Quincea\u00f1era and wedding celebrations, as well as sheet cakes to serve most of the guests. The bakery wants the volume of a small sheet cake to be 351 cubic inches. The cake is in the shape of a rectangular solid. They want the length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of the width. What should the dimensions of the cake pan be?<\/p>\r\n<p id=\"eip-151\">This problem can be solved by writing a cubic function and solving a cubic equation for the volume of the cake. In this section, we will discuss a variety of tools for writing polynomial functions and solving polynomial equations.<\/p>\r\n\r\n<section id=\"fs-id1165135533136\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Evaluating a Polynomial Using the Remainder Theorem<\/h2>\r\n<p id=\"fs-id1165135471230\">In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the <strong>Remainder Theorem<\/strong>. If the polynomial is divided by [latex] x-k, [\/latex] the remainder may be found quickly by evaluating the polynomial function at [latex] k, [\/latex] that is, [latex] f(k). [\/latex] Let\u2019s walk through the proof of the theorem.<\/p>\r\n<p id=\"fs-id1165134085965\">Recall that the Division Algorithm states that, given a polynomial dividend [latex] f(x) [\/latex] and a non-zero polynomial divisor [latex] d(x) [\/latex] there exist unique polynomials [latex] q(x) [\/latex] and [latex] r(x) [\/latex] such that<\/p>\r\n<p style=\"text-align: center;\">[latex] f(x)=d(x)q(x)+r(x) [\/latex]<\/p>\r\n<p id=\"fs-id1165134094600\">and either [latex] r(x)=0 [\/latex] or the degree of [latex] r(x) [\/latex] is less than the degree of [latex] d(x). [\/latex] In practice divisors, [latex] d(x) [\/latex] will have degrees less than or equal to the degree of [latex] f(x). [\/latex] If the divisor, [latex] d(x), [\/latex] is [latex] x-k, [\/latex] this takes the form<\/p>\r\n<p style=\"text-align: center;\">[latex] f(x)=(x-k)(q(x)+r [\/latex]<\/p>\r\n<p id=\"fs-id1165137447771\">Since the divisor [latex] x-k [\/latex] is linear, the remainder will be a constant, [latex] r. [\/latex] And, if we evaluate this for [latex] x=k, [\/latex] we have<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} f(k) &amp;=&amp; (k-k)q(k)+r \\\\ &amp;=&amp; 0\\cdot q(k)+r \\\\ &amp;=&amp; r \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165135572088\">In other words, [latex] f(k) [\/latex] is the remainder obtained by dividing [latex] f(x) [\/latex] by [latex] x-k. [\/latex]<\/p>\r\n\r\n<\/section><\/div>\r\n<div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">The Remainder Theorem<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIf a polynomial [latex] f(x) [\/latex] is divided by [latex] x-k, [\/latex] then the remainder is the value [latex] f(k). [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a polynomial function [latex] f, [\/latex] evaluate [latex] f(x) [\/latex] at [latex] x=k [\/latex] using the Remainder Theorem.<\/strong>\r\n<ol>\r\n \t<li>Use synthetic division to divide the polynomial by [latex] x-k. [\/latex]<\/li>\r\n \t<li>The remainder is the value [latex] f(k). [\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div><\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 1: Using the Remainder Theorem to Evaluate a Polynomial<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse the Remainder Theorem to evaluate [latex] f(x)=6x^4-x^3-15x^2+2x-7 [\/latex] at [latex] x=2. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>To find the remainder using the Remainder Theorem, use synthetic division to divide the polynomial by [latex] x-2. [\/latex]\r\n\r\n<img class=\"size-medium wp-image-853 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/55.-ex-1-300x71.jpeg\" alt=\"\" width=\"300\" height=\"71\" \/>\r\n\r\nThe remainder is 25. Therefore, [latex] f(2)=25. [\/latex]\r\n<h3>Analysis<\/h3>\r\nWe can check our answer by evaluating [latex] f(2). [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} f(x) &amp;=&amp; 6x^4-x^3-15x^2+2x-7 \\\\ f(2) &amp;=&amp; 6(2)^4-(2)^3-15(2)^2+2(2)-7 \\\\ &amp;=&amp; 25 \\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div><\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse the Remainder Theorem to evaluate [latex] f(x)=2x^5-3x^4-9x^3+8x^2+2 [\/latex] at [latex] x=-3. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165137894544\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using the Factor Theorem to Solve a Polynomial Equation<\/h2>\r\n<p id=\"fs-id1165137459796\">The <strong>Factor Theorem <\/strong>is another theorem that helps us analyze polynomial equations. It tells us how the zeros of a polynomial are related to the factors. Recall that the Division Algorithm.<\/p>\r\n<p style=\"text-align: center;\">[latex] f(x)=(x-k)q(x)+r [\/latex]<\/p>\r\n<p id=\"fs-id1165137463592\">If [latex] k [\/latex] is a zero, then the remainder [latex] r [\/latex] is [latex] f(k)=0 [\/latex] and [latex] f(x)=(x-k)q(x)+0 [\/latex] or [latex] f(x)=(x-k)q(x). [\/latex]<\/p>\r\n<p id=\"fs-id1165135176357\">Notice, written in this form, [latex] x-k [\/latex] is a factor of [latex] f(x). [\/latex] We can conclude if [latex] k [\/latex] is a zero of [latex] f(x), [\/latex] then [latex] x-k [\/latex] is a factor of [latex] f(x). [\/latex]<\/p>\r\n<p id=\"fs-id1165135684373\">Similarly, if [latex] x-k [\/latex] is a factor of [latex] f(x), [\/latex] then the remainder of the Division Algorithm [latex] f(x)=(x-k)q(x)+r [\/latex] is 0. This tells us that [latex] k [\/latex] is a zero.<\/p>\r\n<p id=\"fs-id1165132943504\">This pair of implications is the Factor Theorem. As we will soon see, a polynomial of degree [latex] n [\/latex] in the complex number system will have [latex] n [\/latex] zeros. We can use the Factor Theorem to completely factor a polynomial into the product of [latex] n [\/latex] factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">The Factor Theorem<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAccording to the Factor Theorem, [latex] k [\/latex] is a zero of [latex] f(x) [\/latex] if and only if [latex] (x-k) [\/latex] is a factor of [latex] f(x). [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a factor and a third-degree polynomial, use the Factor Theorem to factor the polynomial.<\/strong>\r\n<ol>\r\n \t<li>Use synthetic division to divide the polynomial by [latex] (x-k). [\/latex]<\/li>\r\n \t<li>Confirm that the remainder is 0.<\/li>\r\n \t<li>Write the polynomial as the product of [latex] (x-k) [\/latex] and the quadratic quotient.<\/li>\r\n \t<li>If possible, factor the quadratic.<\/li>\r\n \t<li>Write the polynomial as the product of factors.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div><\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 2: Using the Factor Theorem to Find the Zeros of a Polynomial Expression<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nShow that [latex] (x+2) [\/latex] is a factor of [latex] x^3-6x^2-x+30. [\/latex] Find the remaining factors. Use the factors to determine the zeros of the polynomial.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We can use synthetic division to show that [latex] (x+2) [\/latex] is a factor of the polynomial.\r\n\r\n<img class=\"size-medium wp-image-854 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-2-300x71.jpeg\" alt=\"\" width=\"300\" height=\"71\" \/>\r\n\r\nThe remainder is zero, so [latex] (x+2) [\/latex] is a factor of the polynomial. We can use the Division Algorithm to write the polynomial as the product of the divisor and the quotient:\r\n<p style=\"text-align: center;\">[latex] (x+2)(x^2-8x+15) [\/latex]<\/p>\r\nWe can factor the quadratic factor to write the polynomial as\r\n<p style=\"text-align: center;\">[latex] (x+2)(x-3)(x-5) [\/latex]<\/p>\r\nBy the Factor Theorem, the zeros of [latex] x^3-6x^2-x+30 [\/latex] are [latex] -2, 3 \\ \\ \\text{and} \\ \\ 5. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div><\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse the Factor Theorem to find the zeros of [latex] f(x)=x^3+4x^2-4x-16 [\/latex] given that [latex] (x-2) [\/latex] is a factor of the polynomial.\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165134152972\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using the Rational Zero Theorem to Find Rational Zeros<\/h2>\r\n<p id=\"fs-id1165137660817\">Another use for the Remainder Theorem is to test whether a rational number is a zero for a given polynomial. But first we need a pool of rational numbers to test. The <strong><span id=\"term-00010\" data-type=\"term\">Rational Zero Theorem<\/span><\/strong> helps us to narrow down the number of possible rational zeros using the ratio of the factors of the constant term and factors of the leading <span id=\"term-00011\" class=\"no-emphasis\" data-type=\"term\">coefficient<\/span> of the polynomial<\/p>\r\n<p id=\"fs-id1165135508309\">Consider a quadratic function with two zeros, [latex] x=\\frac{2}{5} [\/latex] and [latex] x=\\frac{3}{4}. [\/latex] By the Factor Theorem, these zeros have factors associated with them. Let us set each factor equal to 0, and then construct the original quadratic function absent its stretching factor.<\/p>\r\n<span id=\"eip-id1165135315549\" data-type=\"media\" data-alt=\"\" data-display=\"block\"><img class=\" wp-image-855 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-rational-zero-300x65.jpeg\" alt=\"\" width=\"420\" height=\"91\" \/><\/span>\r\n<p id=\"fs-id1165135485170\">Notice that two of the factors of the constant term, 6, are the two numerators from the original rational roots: 2 and 3. Similarly, two of the factors from the leading coefficient, 20, are the two denominators from the original rational roots: 5 and 4.<\/p>\r\n<p id=\"fs-id1165137761317\">We can infer that the numerators of the rational roots will always be factors of the constant term and the denominators will be factors of the leading coefficient. This is the essence of the Rational Zero Theorem; it is a means to give us a pool of possible rational zeros.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">The Rational Zero Theorem<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe <span id=\"term-00012\" data-type=\"term\">Rational Zero Theorem<\/span> states that, if the polynomial [latex] f(x)=a_nx^n+a_{n-1}x^{n-1}+\\ldots +a_1x+a_0 [\/latex] has integer coefficients [latex] a_n\\not=0, [\/latex] then every rational zero of [latex] f(x) [\/latex] has the form [latex] \\frac{p}{q} [\/latex] where [latex] p [\/latex] is a factor of the constant term [latex] a_0 [\/latex] and [latex] q [\/latex] is a factor of the leading coefficient [latex] a_n. [\/latex]\r\n\r\nWhen the leading coefficient is 1, the possible rational zeros are the factors of the constant term.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a polynomial function [latex] f(x) [\/latex] use the Rational Zero Theorem to find rational zeros.<\/strong>\r\n<ol>\r\n \t<li>Determine all factors of the constant term and all factors of the leading coefficient.<\/li>\r\n \t<li>Determine all possible values of [latex] \\frac{p}{q}, [\/latex] where [latex] p [\/latex] is a factor of the constant term and [latex] q [\/latex] is a factor of the leading coefficient. Be sure to include both positive and negative candidates.<\/li>\r\n \t<li>Determine which possible zeros are actual zeros by evaluating each case of [latex] f(\\frac{p}{q}). [\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div><\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3: Listing All Possible Rational Zeros<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1165135640968\">List all possible rational zeros of [latex] f(x)=2x^4-5x^3+x^2-4. [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The only possible rational zeros of [latex] f(x) [\/latex] are the quotients of the factors of the last term, [latex] -4, [\/latex] and the factors of the leading coefficient, 2.\r\n\r\nThe constant term is [latex] -4; [\/latex] the factors of [latex] -4 [\/latex] are [latex] p=\\pm1, \\pm2, \\pm4. [\/latex]\r\n\r\nThe leading coefficient is 2; the factors of 2 are [latex] q=\\pm1, \\pm2. [\/latex]\r\n\r\nIf any of the four real zeros are rational zeros, then they will be of one of the following factors of [latex] -4 [\/latex] divided by one of the factors of 2.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{lll} \\frac{p}{q}=\\pm\\frac{1}{1}, \\pm\\frac{1}{2} &amp;&amp; \\frac{p}{q}=\\pm\\frac{2}{1}, \\pm\\frac{2}{2} &amp;&amp; \\frac{p}{q}=\\pm\\frac{4}{1}, \\pm\\frac{4}{2} \\end{array} [\/latex]<\/p>\r\nNote that [latex] \\frac{2}{2}=1 [\/latex] and [latex] \\frac{4}{2}=2, [\/latex] which have already been listed. So we can shorten our list\r\n<p style=\"text-align: center;\">[latex] \\frac{p}{q}=\\frac{\\text{Factors of the last}}{\\text{Factors of the first}}=\\pm1, \\pm2, \\pm4, \\pm\\frac{1}{2} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 4: Using the Rational Zero Theorem to Find Rational Zeros<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse the Rational Zero Theorem to find the rational zeros of [latex] f(x)=2x^3+x^2-4x+1. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The Rational Zero Theorem tells us that if [latex] \\frac{p}{q} [\/latex] is a zero of [latex] f(x),[\/latex] then [latex] p [\/latex] is a factor of 1 and [latex] q [\/latex] is a factor of 2.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} \\frac{p}{q} &amp;=&amp; \\frac{\\text{factor of constant term}}{\\text{factor of leading coefficient}} \\\\ &amp;=&amp; \\frac{\\text{factor of } 1}{\\text{factor of } 2} \\end{array} [\/latex]<\/p>\r\nThe factors of 1 are [latex] \\pm1 [\/latex] and the factors of 2 are [latex] \\pm1 [\/latex] and [latex] \\pm2. [\/latex] The possible values for [latex] \\frac{p}{q} [\/latex] are [latex] \\pm1 [\/latex] and [latex] \\pm\\frac{1}{2}. [\/latex] These are the possible rational zeros for the function. We can determine which of the possible zeros are actual zeros by substituting these values for [latex] x [\/latex] in [latex] f(x). [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} f(-1) &amp;=&amp; 2(-1)^3+(-1)^2-4(-1)+1 &amp;=&amp; 4 \\\\ f(1) &amp;=&amp; 2(1)^3+(1)^2-4(1)+1 &amp;=&amp; 0 \\\\ f(-\\frac{1}{2}) &amp;=&amp; 2(-\\frac{1}{2})^3+(-\\frac{1}{2})^2-4(-\\frac{1}{2})+1 &amp;=&amp; 3 \\\\ f(\\frac{1}{2}) &amp;=&amp; 2(\\frac{1}{2})^3+(\\frac{1}{2})^2-4(\\frac{1}{2})+1 &amp;=&amp; -\\frac{1}{2} \\end{array} [\/latex]<\/p>\r\nOf those, [latex] -1, -\\frac{1}{2}, \\ \\text{and } \\frac{1}{2} [\/latex] are not zeros of [latex] f(x). [\/latex] 1 is the only rational zero of [latex] f(x). [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse the Rational Zero Theorem to find the rational zeros of [latex] f(x)=x^3-5x^2+2x+1. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165137724990\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Finding the Zeros of Polynomial Functions<\/h2>\r\n<p id=\"fs-id1165135530405\">The Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. Once we have done this, we can use synthetic division repeatedly to determine all of the <strong>zeros<\/strong> of a polynomial function.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a polynomial function <\/strong> [latex] f, [\/latex] use synthetic division to find its zeros.\r\n<ol>\r\n \t<li>Use the Rational Zero Theorem to list all possible rational zeros of the function.<\/li>\r\n \t<li>Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the polynomial. If the remainder is 0, the candidate is a zero. If the remainder is not zero, discard the candidate.<\/li>\r\n \t<li>Repeat step two using the quotient found with synthetic division. If possible, continue until the quotient is a quadratic.<\/li>\r\n \t<li>Find the zeros of the quadratic function. Two possible methods for solving quadratics are factoring and using the quadratic formula.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 5: Finding the Zeros of a Polynomial Function with Repeated Real Zeros<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the zeros of [latex] f(x)=4x^3-3x-1. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The Rational Zero Theorem tells us that if [latex] \\frac{p}{q} [\/latex] is a zero of [latex] f(x), [\/latex] then [latex] p [\/latex] is a factor of -1 and [latex] q [\/latex] is a factor of 4.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} \\frac{p}{q} &amp;=&amp; \\frac{\\text{factor of constant term}}{\\text{factor of leading coefficient}} \\\\ &amp;=&amp; \\frac{\\text{factor of } -1}{\\text{factor of } 4} \\end{array} [\/latex]<\/p>\r\nThe factors of [latex] -1 [\/latex] are [latex] \\pm1 [\/latex] and the factors of [latex] 4 [\/latex] are [latex] \\pm1, \\pm2, [\/latex] and [latex] \\pm4. [\/latex] The possible values for [latex] \\frac{p}{q} [\/latex] are [latex] \\pm1, \\pm\\frac{1}{2}, [\/latex] and [latex] \\pm\\frac{1}{4}. [\/latex] These are the possible rational zeros for the function. We will use synthetic division to evaluate each possible zero until we find one that gives a remainder of 0. Let\u2019s begin with 1.\r\n\r\n<img class=\"size-medium wp-image-856 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-5-300x71.jpeg\" alt=\"\" width=\"300\" height=\"71\" \/>\r\n\r\nDividing by [latex] (x-1) [\/latex] gives a remainder of 0, so 1 is a zero of the function. The polynomial can be written as\r\n<p style=\"text-align: center;\">[latex] (x-1)(4x^2+4x+1) [\/latex]<\/p>\r\nThe quadratic is a perfect square. [latex] f(x) [\/latex] can be written as\r\n<p style=\"text-align: center;\">[latex] (x-1)(2x+1)^2 [\/latex]<\/p>\r\nWe already know that 1 is a zero. The other zero will have a multiplicity of 2 because the factor is squared. To find the other zero, we can set the factor equal to 0.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rr} 2x+1 &amp;=&amp; 0 \\\\ x &amp;=&amp; -\\frac{1}{2} \\end{array} [\/latex]<\/p>\r\nThe zeros of the function are 1 and [latex] -\\frac{1}{2} [\/latex] with multiplicity 2.\r\n<h3>Analysis<\/h3>\r\nLook at the graph of the function [latex] f [\/latex] in Figure 2. Notice, at [latex] x=-0.5, [\/latex] the graph bounces off the <em data-effect=\"italics\">x<\/em>-axis, indicating the even multiplicity (2,4,6\u2026) for the zero [latex] -0.5. [\/latex] At [latex] x=1, [\/latex] the graph crosses the <em data-effect=\"italics\">x<\/em>-axis, indicating the odd multiplicity (1,3,5\u2026) for the zero [latex] x=1. [\/latex]\r\n\r\n[caption id=\"attachment_857\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-857\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-1-300x178.jpeg\" alt=\"\" width=\"300\" height=\"178\" \/> Figure 2[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div><section id=\"fs-id1165137461780\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using the Fundamental Theorem of Algebra<\/h2>\r\n<p id=\"fs-id1165135547255\">Now that we can find rational zeros for a polynomial function, we will look at a theorem that discusses the number of complex zeros of a polynomial function. The <strong>Fundamental Theorem of Algebra <\/strong>tells us that every polynomial function has at least one complex zero. This theorem forms the foundation for solving polynomial equations.<\/p>\r\n<p id=\"fs-id1165137771429\">Suppose [latex] f [\/latex] is a polynomial function of degree four, and [latex] f(x)=0. [\/latex] The Fundamental Theorem of Algebra states that there is at least one complex solution, call it [latex] c_1. [\/latex] By the Factor Theorem, we can write [latex] f(x) [\/latex] as a product of [latex] x-c_1 [\/latex] and a polynomial quotient. Since [latex] x-c_1 [\/latex] is linear, the polynomial quotient will be of degree three. Now we apply the Fundamental Theorem of Algebra to the third-degree polynomial quotient. It will have at least one complex zero, call it [latex] c_2. [\/latex] So we can write the polynomial quotient as a product of [latex] x-c_2 [\/latex] and a new polynomial quotient of degree two. Continue to apply the Fundamental Theorem of Algebra until all of the zeros are found. There will be four of them and each one will yield a factor of [latex] f(x). [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">The Fundamental Theorem of Algebra<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe <strong><span id=\"term-00015\" data-type=\"term\">Fundamental Theorem of Algebra<\/span><\/strong> states that, if [latex] f(x) [\/latex] is a polynomial of degree [latex] n&gt; 0, [\/latex] then [latex] f(x) [\/latex] has at least one complex zero.\r\n\r\nWe can use this theorem to argue that, if [latex] f(x) [\/latex] is a polynomial of degree [latex] n&gt; 0, [\/latex] and [latex] a [\/latex] is a non-zero real number, then [latex] f(x) [\/latex] has exactly [latex] n [\/latex] linear factors\r\n<p style=\"text-align: center;\">[latex] f(x)=a(x-c_1)(x-c_2)\\ldots(x-c_n) [\/latex]<\/p>\r\nwhere [latex] c_1, c_2, \\ldots, c_n [\/latex] are complex numbers. Therefore, [latex] f(x) [\/latex] has [latex] n [\/latex] roots if we allow for multiplicities.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div>\r\n<div class=\"os-note-body\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Q: Does every polynomial have at least one imaginary zero?<\/strong>\r\n\r\nA: <em data-effect=\"italics\">No. Real numbers are a subset of complex numbers, but not the other way around. A complex number is not necessarily imaginary. Real numbers are also complex numbers.<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 6: Finding the Zeros of a Polynomial Function with Complex Zeros<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the zeros of [latex] f(x)=3x^3+9x^2+x+3. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The Rational Zero Theorem tells us that if [latex] \\frac{p}{q} [\/latex] is a zero of [latex] f(x), [\/latex] then [latex] p [\/latex] is a factor of 3 and [latex] q [\/latex] is a factor of 3.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} \\frac{p}{q} &amp;=&amp; \\frac{\\text{factor of constant term}}{\\text{factor of leading coefficient}} \\\\ &amp;=&amp; \\frac{\\text{factor of } 3}{\\text{factor of } 3} \\end{array} [\/latex]<\/p>\r\nThe factors of 3 are [latex] \\pm1 [\/latex] and [latex] \\pm3. [\/latex] The possible values for [latex] \\frac{p}{q} [\/latex] and therefore the possible rational zeros for the function, are [latex] \\pm3, \\pm1, \\ \\text{and } \\pm\\frac{1}{3}. [\/latex] We will use synthetic division to evaluate each possible zero until we find one that gives a remainder of 0. Let\u2019s begin with \u20133.\r\n\r\n<img class=\"size-medium wp-image-858 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-6-300x71.jpeg\" alt=\"\" width=\"300\" height=\"71\" \/>\r\n\r\nDividing by [latex] (x+3) [\/latex] gives a remainder of 0, so [latex] -3 [\/latex] is a zero of the function. The polynomial can be written as\r\n<p style=\"text-align: center;\">[latex] (x+3)(3x^2+1) [\/latex]<\/p>\r\nWe can then set the quadratic equal to 0 and solve to find the other zeros of the function.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rr} 3x^2+1 &amp;=&amp; 0 \\\\ x^2 &amp;=&amp; -\\frac{1}{3} \\\\ x &amp;=&amp; \\pm\\sqrt{-\\frac{1}{3}} &amp;=&amp; \\pm\\frac{i\\sqrt{3}}{3} \\end{array} [\/latex]<\/p>\r\nThe zeros [latex] f(x) [\/latex] of are [latex] -3 [\/latex] and [latex] \\pm\\frac{i\\sqrt{3}}{3}. [\/latex]\r\n<h3>Analysis<\/h3>\r\nLook at the graph of the function [latex] f [\/latex] in Figure 3. Notice that, at [latex] x=-3, [\/latex] the graph crosses the <em data-effect=\"italics\">x<\/em>-axis, indicating an odd multiplicity (1) for the zero [latex] x=-3. [\/latex] Also note the presence of the two turning points. This means that, since there is a 3<sup>rd<\/sup> degree polynomial, we are looking at the maximum number of turning points. So, the end behavior of increasing without bound to the right and decreasing without bound to the left will continue. Thus, all the <em data-effect=\"italics\">x<\/em>-intercepts for the function are shown. So either the multiplicity of [latex] x=-3 [\/latex] is 1 and there are two complex solutions, which is what we found, or the multiplicity at [latex] x=-3 [\/latex] is three. Either way, our result is correct.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_859\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-859\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-2-300x178.jpeg\" alt=\"\" width=\"300\" height=\"178\" \/> Figure 3[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<section><\/section><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the zeros of [latex] f(x)=2x^3+5x^2-11x+4. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165135501998\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using the Linear Factorization Theorem to Find Polynomials with Given Zeros<\/h2>\r\n<p id=\"fs-id1165135502003\">A vital implication of the <span id=\"term-00016\" class=\"no-emphasis\" data-type=\"term\">Fundamental Theorem of Algebra<\/span>, as we stated above, is that a polynomial function of degree [latex] n [\/latex] will have [latex] n [\/latex] zeros in the set of complex numbers, if we allow for multiplicities. This means that we can factor the polynomial function into [latex] n [\/latex] factors. The<strong> Linear Factorization Theorem<\/strong> tells us that a polynomial function will have the same number of factors as its degree, and that each factor will be in the form [latex] (x-c), [\/latex] where [latex] c [\/latex] is a complex number.<\/p>\r\n<p id=\"eip-651\">Let [latex] f [\/latex] be a polynomial function with real coefficients, and suppose [latex] a+bi, b\\not=0, [\/latex] is a zero of [latex] f(x). [\/latex] Then, by the Factor Theorem, [latex] x-(a+bi) [\/latex] is a factor of [latex] f(x). [\/latex] For [latex] f [\/latex] to have real coefficients, [latex] x-(a-bi) [\/latex] must also be a factor of [latex] f(x). [\/latex] This is true because any factor other than [latex] x-(a-bi) [\/latex] when multiplied by [latex] x-(a+bi), [\/latex] will leave imaginary components in the product. Only multiplication with conjugate pairs will eliminate the imaginary parts and result in real coefficients. In other words, if a polynomial function [latex] f [\/latex] with real coefficients has a complex zero [latex] a+bi, [\/latex] then the complex conjugate [latex] a-bi [\/latex] must also be a zero of [latex] f(x). [\/latex] This is called the Complex Conjugate Theorem.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Complex Conjugate Theorem<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAccording to the <strong>Linear Factorization Theorem<\/strong>, a polynomial function will have the same number of factors as its degree, and each factor will be in the form [latex] (x-c), [\/latex] where [latex] c [\/latex] is a complex number.\r\n\r\nIf the polynomial function [latex] f [\/latex] has real coefficients and a complex zero in the form [latex] a+bi, [\/latex] then the complex conjugate of the zero, [latex] a-bi, [\/latex] is also a zero.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given the zeros of a polynomial function [latex] f [\/latex] and a point [latex] (c, f(c)) [\/latex] on the graph of [latex] f, [\/latex] use the Linear Factorization Theorem to find the polynomial function.<\/strong>\r\n<ol>\r\n \t<li>Use the zeros to construct the linear factors of the polynomial.<\/li>\r\n \t<li>Multiply the linear factors to expand the polynomial.<\/li>\r\n \t<li>Substitute [latex] (c, f(c)) [\/latex] into the function to determine the leading coefficient<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 7: Using the Linear Factorization Theorem to Find a Polynomial with Given Zeros<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind a fourth degree polynomial with real coefficients that has zeros of [latex] -3, 2, i, [\/latex] such that [latex] f(-2)=100. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Because [latex] x=i [\/latex] is a zero, by the Complex Conjugate Theorem [latex] x=-i [\/latex] is also a zero. The polynomial must have factors of [latex] (x+3), (x-2), (x-i), [\/latex] and [latex] (x+i). [\/latex] Since we are looking for a degree 4 polynomial, and now have four zeros, we have all four factors. Let\u2019s begin by multiplying these factors.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} f(x) &amp;=&amp; a(x+3)(x-2)(x-i)(x+i) \\\\ f(x) &amp;=&amp; a(x^2+x-6)(x^2+1) \\\\ f(x) &amp;=&amp; a(x^4+x^3-5x^2+x-6) \\end{array} [\/latex]<\/p>\r\nWe need to find [latex] a [\/latex] to ensure [latex] f(-2)=100. [\/latex] Substitute [latex] x=-2 [\/latex] and [latex] f(2)=100 [\/latex] into [latex] f(x). [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} 100 &amp;=&amp; a((-2)^4+(-2)^3-5(-2)^2+(-2)-6) \\\\ 100 &amp;=&amp; a(-2) \\\\ -5 &amp;=&amp; a \\end{array} [\/latex]<\/p>\r\nSo the polynomial function is\r\n<p style=\"text-align: center;\">[latex] f(x)=-5(x^4+x^3-5x^2+x-6) [\/latex]<\/p>\r\nor\r\n<p style=\"text-align: center;\">[latex] f(x)=-5x^4-5x^3+25x^2-5x+30 [\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nWe found that both [latex] i [\/latex] and [latex] -i [\/latex] were zeros, but only one of these zeros needed to be given. If [latex] i [\/latex] is a zero of a polynomial with real coefficients, then [latex] -i [\/latex] must also be a zero of the polynomial because is the complex conjugate of [latex] i. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<section><\/section><\/div>\r\n<div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Q: If <\/strong> [latex] 2+3i [\/latex] were given as a zero of a polynomial with real coefficients, would [latex] 2-3i [\/latex] also need to be a zero?\r\n\r\nA: <em data-effect=\"italics\">Yes. When any complex number with an imaginary component is given as a zero of a polynomial with real coefficients, the conjugate must also be a zero of the polynomial.<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<section><\/section><\/div>\r\n<div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind a third degree polynomial with real coefficients that has zeros of 5 and [latex] -2i [\/latex] such that [latex] f(1)=10. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165135177650\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using Descartes\u2019 Rule of Signs<\/h2>\r\n<p id=\"fs-id1165135177655\">There is a straightforward way to determine the possible numbers of positive and negative real zeros for any polynomial function. If the polynomial is written in descending order,<strong> Descartes\u2019 Rule of Signs<\/strong> tells us of a relationship between the number of sign changes in [latex] f(x) [\/latex] and the number of positive real zeros. For example, the polynomial function below has one sign change.<span id=\"fs-id1165134378690\" data-type=\"media\" data-alt=\"The function, f(x)=x^4+x^3+x^2+x-1, has one sign change between x and -1.&#96;\" data-display=\"block\"><\/span><\/p>\r\n<img class=\" wp-image-1560 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-300x28.webp\" alt=\"\" width=\"321\" height=\"30\" \/>\r\n<p id=\"fs-id1165135206081\">This tells us that the function must have 1 positive real zero.<\/p>\r\n<p id=\"fs-id1165135206084\">There is a similar relationship between the number of sign changes in [latex] f(-x) [\/latex] and the number of negative real zeros.<span id=\"fs-id1165135152070\" data-type=\"media\" data-alt=\"The function, f(-x)=(-x)^4+(-x)^3+(-x)^2+(-x)-1=+ x^4-x^3+x^2-x-1, has three sign changes between x^4 and x^3, x^3 and x^2, and x^2 and x.&#96;\" data-display=\"block\"><\/span><\/p>\r\n<img class=\" wp-image-1561 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5.1-300x47.webp\" alt=\"\" width=\"345\" height=\"54\" \/>\r\n<p id=\"fs-id1165135152083\">In this case, [latex] f(-x) [\/latex] has 3 sign changes. This tells us that [latex] f(x) [\/latex] could have 3 or 1 negative real zeros.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Descartes' Rule of Signs<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAccording to <strong>Descartes\u2019 Rule of Signs<\/strong>, if we let [latex] f(x)=a_nx^n+a_{n-1}x^{n-1}+\\ldots+a_1x+a_0 [\/latex] be a polynomial function with real coefficients:\r\n<ul>\r\n \t<li>The number of positive real zeros is either equal to the number of sign changes of [latex] f(x) [\/latex] or is less than the number of sign changes by an even integer.<\/li>\r\n \t<li>The number of negative real zeros is either equal to the number of sign changes of [latex] f(-x) [\/latex] or is less than the number of sign changes by an even integer.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 8: Using Descartes' Rule of Signs<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse Descartes\u2019 Rule of Signs to determine the possible numbers of positive and negative real zeros for [latex] f(x)=-x^4-3x^3+6x^2-4x-12. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Begin by determining the number of sign changes.\r\n\r\n[caption id=\"attachment_860\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-860\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-3-300x32.jpeg\" alt=\"\" width=\"300\" height=\"32\" \/> Figure 4[\/caption]\r\n\r\nThere are two sign changes, so there are either 2 or 0 positive real roots. Next, we examine [latex] f(-x) [\/latex] to determine the number of negative real roots.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} f(-x) &amp;=&amp; -(-x)^4--3(-x)^3+6(-x)^2-4(-x)-12 \\\\ f(-x) &amp;=&amp; -x^4+3x^3+6x^2+4x-12 \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_861\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-861\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-4-300x30.jpeg\" alt=\"\" width=\"300\" height=\"30\" \/> Figure 5[\/caption]\r\n\r\nAgain, there are two sign changes, so there are either 2 or 0 negative real roots.\r\n\r\nThere are four possibilities, as we can see in Table 1.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 1<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 25%; text-align: center;\"><strong>Positive Real Zeros<\/strong><\/td>\r\n<td style=\"width: 25%; text-align: center;\"><strong>Negative Real Zeros<\/strong><\/td>\r\n<td style=\"width: 25%; text-align: center;\"><strong>Complex Zeros<\/strong><\/td>\r\n<td style=\"width: 25%; text-align: center;\"><strong>Total Zeros<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%; text-align: center;\">2<\/td>\r\n<td style=\"width: 25%; text-align: center;\">2<\/td>\r\n<td style=\"width: 25%; text-align: center;\">0<\/td>\r\n<td style=\"width: 25%; text-align: center;\">4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%; text-align: center;\">2<\/td>\r\n<td style=\"width: 25%; text-align: center;\">0<\/td>\r\n<td style=\"width: 25%; text-align: center;\">2<\/td>\r\n<td style=\"width: 25%; text-align: center;\">4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%; text-align: center;\">0<\/td>\r\n<td style=\"width: 25%; text-align: center;\">2<\/td>\r\n<td style=\"width: 25%; text-align: center;\">2<\/td>\r\n<td style=\"width: 25%; text-align: center;\">4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%; text-align: center;\">0<\/td>\r\n<td style=\"width: 25%; text-align: center;\">0<\/td>\r\n<td style=\"width: 25%; text-align: center;\">4<\/td>\r\n<td style=\"width: 25%; text-align: center;\">4<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3>Analysis<\/h3>\r\nWe can confirm the numbers of positive and negative real roots by examining a graph of the function. See Figure 6. We can see from the graph that the function has 0 positive real roots and 2 negative real roots.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_862\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-862\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-5-300x153.jpeg\" alt=\"\" width=\"300\" height=\"153\" \/> Figure 6[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<section><\/section><\/div>\r\n<div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse Descartes\u2019 Rule of Signs to determine the maximum possible numbers of positive and negative real zeros for [latex] f(x)=2x^4-10x^3+11x^2-15x+12. [\/latex] Use a graph to verify the numbers of positive and negative real zeros for the function.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165135440213\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Solving Real-World Applications<\/h2>\r\n<p id=\"fs-id1165135440219\">We have now introduced a variety of tools for solving polynomial equations. Let\u2019s use these tools to solve the bakery problem from the beginning of the section.<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 9: Solving Polynomial Equations<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA new bakery offers decorated, multi-tiered cakes for display and cutting at Quincea\u00f1era and wedding celebrations, as well as sheet cakes to serve most of the guests. The bakery wants the volume of a small sheet cake to be 351 cubic inches. The cake is in the shape of a rectangular solid. They want the length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of the width. What should the dimensions of the cake pan be?\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Begin by writing an equation for the volume of the cake. The volume of a rectangular solid is given by [latex] V=lwh. [\/latex] We were given that the length must be four inches longer than the width, so we can express the length of the cake as [latex] l=w+4. [\/latex] We were given that the height of the cake is one-third of the width, so we can express the height of the cake as [latex] h=\\frac{1}{3}w. [\/latex] Let\u2019s write the volume of the cake in terms of width of the cake.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} V &amp;=&amp; (w+4)(w)(\\frac{1}{3}w) \\\\ V &amp;=&amp; \\frac{1}{3}w^3+\\frac{4}{3}w^2 \\end{array} [\/latex]<\/p>\r\nSubstitute the given volume into this equation.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} 351 &amp;=&amp; \\frac{1}{3}w^3+\\frac{4}{3}w^2 &amp; \\quad \\text{Substitue 351 for } V. \\\\ 1053 &amp;=&amp; w^3+4w^2 &amp; \\quad \\text{Multiply both sides by 3.} \\\\ 0 &amp;=&amp; w^3+4w^2-1053 &amp; \\quad \\text{Subtract 1-53 from both sides.} \\end{array} [\/latex]<\/p>\r\nDescartes' rule of signs tells us there is one positive solution. The Rational Zero Theorem tells us that the possible rational zeros are [latex] \\pm1, \\pm3, \\pm9, \\pm13, \\pm27, \\pm39, \\pm81, \\pm117, \\pm351 [\/latex] and [latex] \\pm1053. [\/latex] We can use synthetic division to test these possible zeros. Only positive numbers make sense as dimensions for a cake, so we need not test any negative values. Let\u2019s begin by testing values that make the most sense as dimensions for a small sheet cake. Use synthetic division to check [latex] x=1. [\/latex]\r\n\r\n<img class=\"size-medium wp-image-863 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-9.1-300x71.jpeg\" alt=\"\" width=\"300\" height=\"71\" \/>\r\n\r\nSince 1 is not a solution, we will check [latex] x=3. [\/latex]\r\n\r\n<img class=\"size-medium wp-image-864 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-9.2-300x71.jpeg\" alt=\"\" width=\"300\" height=\"71\" \/>\r\n\r\nSince 3 is not a solution either, we will test [latex] x=9. [\/latex]\r\n\r\n<img class=\"size-medium wp-image-865 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-9.3-300x71.jpeg\" alt=\"\" width=\"300\" height=\"71\" \/>\r\n\r\nSynthetic division gives a remainder of 0, so 9 is a solution to the equation. We can use the relationships between the width and the other dimensions to determine the length and height of the sheet cake pan.\r\n<p style=\"text-align: center;\">[latex] l=w+4=9+4=13 \\ \\text{and } h=\\frac{1}{3}w=\\frac{1}{3}(9)=3 [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA shipping container in the shape of a rectangular solid must have a volume of 84 cubic meters. The client tells the manufacturer that, because of the contents, the length of the container must be one meter longer than the width, and the height must be one meter greater than twice the width. What should the dimensions of the container be?\r\n\r\n<\/div>\r\n<\/div>\r\n<section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_03_06_07\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137843250\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Media<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAccess these online resources for additional instruction and practice with zeros of polynomial functions.\r\n<ul>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=e_EttLeQblY\">Real Zeros, Factors, and Graphs of Polynomial Functions<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=F0bcRJfdh5Q\">Complex Factorization Theorem<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=LeZdCSCIb3Q\">Find the Zeros of a Polynomial Function<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=D_I11k2DfCg\">Find the Zeros of a Polynomial Function 2<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=paq5VWwXHp8\">Find the Zeros of a Polynomial Function 3<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\r\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">5.5 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1165135149755\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1165135149759\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1165135149765\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135149766\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135149765-solution\">1<\/a><span class=\"os-divider\">. <\/span>Describe a use for the Remainder Theorem.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135149774\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137898784\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>Explain why the Rational Zero Theorem does not guarantee finding zeros of a polynomial function.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137898789\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137898790\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137898789-solution\">3<\/a><span class=\"os-divider\">. <\/span>What is the difference between rational and real zeros?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137898800\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137898801\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>If Descartes\u2019 Rule of Signs reveals a no change of signs or one sign of changes, what specific conclusion can be drawn?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137898807\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137898808\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137898807-solution\">5<\/a><span class=\"os-divider\">. <\/span>If synthetic division reveals a zero, why should we try that value again as a possible solution?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165135531376\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id1165135531382\">For the following exercises, use the Remainder Theorem to find the remainder.<\/p>\r\n\r\n<div id=\"fs-id1165135531385\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135531386\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex] (x^4-9x^2+14)\\div (x-2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>\r\n<div id=\"fs-id1165134152708\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134152709\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134152708-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex] (3x^3=2x^2+x-4)\\div (x+3) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135532325\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135532326\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex] (x^4+5x^3-4x-17)\\div (x+1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135571875\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135571876\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135571875-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex] (-3x^2+6x+24)\\div (x-4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134106023\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137675202\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex] (5x^5-4x^4+3x^3-2x^2+x-1)\\div (x+6) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135487281\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135487282\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135487281-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex] (x^4-1)\\div (x-4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137933131\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137933132\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex] (3x^3+4x^2-8x+2)\\div (x-3) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134430485\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134430486\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134430485-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex] (4x^3+5x^2-2x+7)\\div (x+2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135403423\">For the following exercises, use the given factor and the Factor Theorem to find all real zeros for the given polynomial function.<\/p>\r\n\r\n<div id=\"fs-id1165135403427\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135403428\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=2x^3-9x^2+13x-6; x-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135349107\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135349108\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135349107-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=2x^3+x^2-5x+2; x+2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135198461\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135198462\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=3x^3+x^2-20x+12; x+3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137646943\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137646944\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137646943-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=2x^3+3x^2+x+6; x+2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135363964\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135363965\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=-5x^3+16x^2-9; x-3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137833870\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137833871\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137833870-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=x^3+3x^2+4x+12; x+3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134039305\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134039306\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=4x^3-7x+3; x-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135453082\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135453083\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135453082-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=2x^3+5x^2-12x-30; 2x+5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165134472281\">For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.<\/p>\r\n\r\n<div id=\"fs-id1165134472284\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134472285\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex] x^3-3x^2-10x+24=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134212161\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134212162\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134212161-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] 2x^3+7x^2-10x-24=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135678616\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135678617\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex] x^3+2x^2-9x-18=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135554358\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135554359\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135554358-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex] x^3+5x^2-16x-80=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135154364\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135154365\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex] x^3-3x^2-25x+75=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135389007\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133093326\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135389007-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex] 2x^3-3x^2-32x-15=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137676884\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137676885\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex] 2x^3+x^2-7x-6=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135393418\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135393419\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135393418-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex] 2x^3-3x^2-x+1=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137766744\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137766746\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex] 3x^3-x^2-11x-6=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135356524\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135632075\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135356524-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex] 2x^3-5x^2+9x-9=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135695187\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135695188\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex] 2x^3-3x^2+4x+3=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135692865\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135692866\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135692865-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex] x^4-2x^3-7x^2+8x+12=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134474183\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134474184\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex] x^4+2x^3-9x^2-2x+8=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135443739\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135443740\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135443739-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex] 4x^4+4x^3-25x^2-x+6=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137680607\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137680608\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex] 2x^4-3x^3-15x^2+32x-12=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135353111\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135353112\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135353111-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex] x^4+2x^3-4x^2-10x-5=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133311066\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133311068\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex] 4x^3-3x+1=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133214917\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133214918\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165133214917-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex] 8x^4+26x^3+39x^2+26x+6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135517156\">For the following exercises, find all complex solutions (real and non-real).<\/p>\r\n\r\n<div id=\"fs-id1165135517160\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135517161\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex] x^3+x62+x+1=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132940010\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132940011\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165132940010-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex] x^3-8x^2+25x-26=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135199514\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135199515\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex] x^3+13x^2+57x+85=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134380350\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134380351\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134380350-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex] 3x^3-4x^2+11x+10=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135416512\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135416513\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex] x^4+2x^3+22x^2+50x-75=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137851467\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137851468\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137851467-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex] 2x^3-3x^2+32x+17=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165135403510\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphical<\/h3>\r\n<p id=\"fs-id1165135403515\">For the following exercises, use Descartes\u2019 Rule to determine the possible number of positive and negative solutions. Confirm with the given graph.<\/p>\r\n\r\n<div id=\"fs-id1165135403521\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135403522\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=x^3-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165134061970\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134061972\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134061970-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=x^4-x^2-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135333211\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135333212\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=x^3-2x^2-5x+6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135485740\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135485741\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135485740-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=x^3-2x^2+x-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134261871\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134261872\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=x^4+2x^3-12x^2+14x-5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134149873\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134149874\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134149873-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=2x^3+37x^2+200x+300 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134389900\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134389901\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=x^3-2x^2-16x+32 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134231463\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134231464\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134231463-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=2x^4-5x^3-5x^2+5x+3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134148467\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134148468\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=2x^4-5x^3-14x^2+20x+8 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135640945\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135640946\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135640945-solution\">55<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=10x^4-21x^2+11 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165134058383\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Numeric<\/h3>\r\n<p id=\"fs-id1165134058388\">For the following exercises, list all possible rational zeros for the functions.<\/p>\r\n\r\n<div id=\"fs-id1165134058391\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134058392\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=x^4+3x^3-4x+4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165135486016\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135486017\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135486016-solution\">57<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=2x^3+3x^2-8x+5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134204451\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134204452\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=3x^3+5x^2-5x+4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132920307\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135179860\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165132920307-solution\">59<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=6x^4-10x^2+13x+1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135698642\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135698643\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=4x^5-10x^4+8x^3+x^2-8 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165135353045\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<p id=\"fs-id1165135353050\">For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.<\/p>\r\n\r\n<div id=\"fs-id1165135353055\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135353056\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135353055-solution\">61<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=6x^3-7x^2+1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165134431750\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134431751\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=4x^3-4x^2-13x-5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135394055\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135394056\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135394055-solution\">63<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=8x^3-6x^2-23x+6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135414275\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135414276\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=12x^4+55x^3+12x^2-177x+54 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134469920\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134469921\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134469920-solution\">65<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=16x^4-24x^3+x^2-15x+25 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165134155157\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<p id=\"fs-id1165134155162\">For the following exercises, construct a polynomial function of least degree possible using the given information.<\/p>\r\n\r\n<div id=\"fs-id1165134155167\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134155168\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>Real roots: [latex] -1, 1, 3, \\ \\text{and } (2, f(2))=(2, 4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135538724\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135538725\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135538724-solution\">67<\/a><span class=\"os-divider\">. <\/span>Real roots: [latex] -1, 1 \\ \\text{(with multiplicity 2 and 1) and } (2, f(2))=(2, 4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133213938\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133213939\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span>Real roots: [latex] ,2 \\frac{1}{2} \\ \\text{(with multiplicity 2) and } (-3, f(-3))=(-3, 5) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135205845\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135205846\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135205845-solution\">69<\/a><span class=\"os-divider\">. <\/span>Real roots: [latex] -\\frac{1}{2}, 0, \\frac{1}{2} \\ \\text{and } (-2, f(-2))=(-2, 6) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134047574\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134047575\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">70<\/span><span class=\"os-divider\">. <\/span>Real roots: [latex] -4, -1, 1, 4 \\ \\text{and } (-2, f(-2))=(-2, 10) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165134109613\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<p id=\"fs-id1165134109618\">For the following exercises, find the dimensions of the box described.<\/p>\r\n\r\n<div id=\"fs-id1165134043608\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134043610\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134043608-solution\">71<\/a><span class=\"os-divider\">. <\/span>The length is twice as long as the width. The height is 2 inches greater than the width. The volume is 192 cubic inches.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134043619\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134043620\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">72<\/span><span class=\"os-divider\">. <\/span>The length, width, and height are consecutive whole numbers. The volume is 120 cubic inches.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134043625\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134043626\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134043625-solution\">73<\/a><span class=\"os-divider\">. <\/span>The length is one inch more than the width, which is one inch more than the height. The volume is 86.625 cubic inches.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134043635\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134043636\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">74<\/span><span class=\"os-divider\">. <\/span>The length is three times the height and the height is one inch less than the width. The volume is 108 cubic inches.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134043642\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134043643\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134043642-solution\">75<\/a><span class=\"os-divider\">. <\/span>The length is 3 inches more than the width. The width is 2 inches more than the height. The volume is 120 cubic inches.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165134043653\">For the following exercises, find the dimensions of the right circular cylinder described.<\/p>\r\n\r\n<div id=\"fs-id1165134043657\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134043658\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">76<\/span><span class=\"os-divider\">. <\/span>The radius is 3 inches more than the height. The volume is [latex] 16\\pi [\/latex] cubic meters.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135315485\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135315486\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135315485-solution\">77<\/a><span class=\"os-divider\">. <\/span>The height is one less than one half the radius. The volume is [latex] 72\\pi [\/latex] cubic meters.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135315510\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135315511\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">78<\/span><span class=\"os-divider\">. <\/span>The radius and height differ by one meter. The radius is larger and the volume is [latex] 48\\pi [\/latex] cubic meters.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135453003\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135453004\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135453003-solution\">79<\/a><span class=\"os-divider\">. <\/span>The radius and height differ by two meters. The height is greater and the volume is [latex] 28.125\\pi [\/latex] cubic meters.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135453028\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135453029\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">80<\/span><span class=\"os-divider\">. <\/span>The radius is [latex] \\frac{1}{3} [\/latex] meter greater than the height. The volume is [latex] \\frac{98}{9}\\pi [\/latex] cubic meters.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_1bb57d2c-789f-4252-bfe4-edc0873f1e29\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>In this section, you will:<\/p>\n<ul>\n<li>Evaluate a polynomial using the Remainder Theorem.<\/li>\n<li>Use the Factor Theorem to solve a polynomial equation.<\/li>\n<li>Use the Rational Zero Theorem to find rational zeros.<\/li>\n<li>Find zeros of a polynomial function.<\/li>\n<li>Use the Linear Factorization Theorem to find polynomials with given zeros.<\/li>\n<li>Use Descartes\u2019 Rule of Signs.<\/li>\n<li>Solve real-world applications of polynomial equations<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<figure id=\"attachment_1551\" aria-describedby=\"caption-attachment-1551\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1551\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-1-Quinceanera-sheet-cake-300x210.jpg\" alt=\"\" width=\"300\" height=\"210\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-1-Quinceanera-sheet-cake-300x210.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-1-Quinceanera-sheet-cake-65x45.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-1-Quinceanera-sheet-cake-225x157.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-1-Quinceanera-sheet-cake-350x245.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-1-Quinceanera-sheet-cake.jpg 679w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1551\" class=\"wp-caption-text\">Figure 1<\/figcaption><\/figure>\n<p id=\"fs-id1165137758829\">A new bakery offers decorated, multi-tiered cakes for display and cutting at Quincea\u00f1era and wedding celebrations, as well as sheet cakes to serve most of the guests. The bakery wants the volume of a small sheet cake to be 351 cubic inches. The cake is in the shape of a rectangular solid. They want the length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of the width. What should the dimensions of the cake pan be?<\/p>\n<p id=\"eip-151\">This problem can be solved by writing a cubic function and solving a cubic equation for the volume of the cake. In this section, we will discuss a variety of tools for writing polynomial functions and solving polynomial equations.<\/p>\n<section id=\"fs-id1165135533136\" data-depth=\"1\">\n<h2 data-type=\"title\">Evaluating a Polynomial Using the Remainder Theorem<\/h2>\n<p id=\"fs-id1165135471230\">In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the <strong>Remainder Theorem<\/strong>. If the polynomial is divided by [latex]x-k,[\/latex] the remainder may be found quickly by evaluating the polynomial function at [latex]k,[\/latex] that is, [latex]f(k).[\/latex] Let\u2019s walk through the proof of the theorem.<\/p>\n<p id=\"fs-id1165134085965\">Recall that the Division Algorithm states that, given a polynomial dividend [latex]f(x)[\/latex] and a non-zero polynomial divisor [latex]d(x)[\/latex] there exist unique polynomials [latex]q(x)[\/latex] and [latex]r(x)[\/latex] such that<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=d(x)q(x)+r(x)[\/latex]<\/p>\n<p id=\"fs-id1165134094600\">and either [latex]r(x)=0[\/latex] or the degree of [latex]r(x)[\/latex] is less than the degree of [latex]d(x).[\/latex] In practice divisors, [latex]d(x)[\/latex] will have degrees less than or equal to the degree of [latex]f(x).[\/latex] If the divisor, [latex]d(x),[\/latex] is [latex]x-k,[\/latex] this takes the form<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=(x-k)(q(x)+r[\/latex]<\/p>\n<p id=\"fs-id1165137447771\">Since the divisor [latex]x-k[\/latex] is linear, the remainder will be a constant, [latex]r.[\/latex] And, if we evaluate this for [latex]x=k,[\/latex] we have<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} f(k) &=& (k-k)q(k)+r \\\\ &=& 0\\cdot q(k)+r \\\\ &=& r \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165135572088\">In other words, [latex]f(k)[\/latex] is the remainder obtained by dividing [latex]f(x)[\/latex] by [latex]x-k.[\/latex]<\/p>\n<\/section>\n<\/div>\n<div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">The Remainder Theorem<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>If a polynomial [latex]f(x)[\/latex] is divided by [latex]x-k,[\/latex] then the remainder is the value [latex]f(k).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a polynomial function [latex]f,[\/latex] evaluate [latex]f(x)[\/latex] at [latex]x=k[\/latex] using the Remainder Theorem.<\/strong><\/p>\n<ol>\n<li>Use synthetic division to divide the polynomial by [latex]x-k.[\/latex]<\/li>\n<li>The remainder is the value [latex]f(k).[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div><\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1: Using the Remainder Theorem to Evaluate a Polynomial<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use the Remainder Theorem to evaluate [latex]f(x)=6x^4-x^3-15x^2+2x-7[\/latex] at [latex]x=2.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>To find the remainder using the Remainder Theorem, use synthetic division to divide the polynomial by [latex]x-2.[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-853 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/55.-ex-1-300x71.jpeg\" alt=\"\" width=\"300\" height=\"71\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/55.-ex-1-300x71.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/55.-ex-1-65x15.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/55.-ex-1-225x53.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/55.-ex-1.jpeg 325w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>The remainder is 25. Therefore, [latex]f(2)=25.[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>We can check our answer by evaluating [latex]f(2).[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} f(x) &=& 6x^4-x^3-15x^2+2x-7 \\\\ f(2) &=& 6(2)^4-(2)^3-15(2)^2+2(2)-7 \\\\ &=& 25 \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div><\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use the Remainder Theorem to evaluate [latex]f(x)=2x^5-3x^4-9x^3+8x^2+2[\/latex] at [latex]x=-3.[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165137894544\" data-depth=\"1\">\n<h2 data-type=\"title\">Using the Factor Theorem to Solve a Polynomial Equation<\/h2>\n<p id=\"fs-id1165137459796\">The <strong>Factor Theorem <\/strong>is another theorem that helps us analyze polynomial equations. It tells us how the zeros of a polynomial are related to the factors. Recall that the Division Algorithm.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=(x-k)q(x)+r[\/latex]<\/p>\n<p id=\"fs-id1165137463592\">If [latex]k[\/latex] is a zero, then the remainder [latex]r[\/latex] is [latex]f(k)=0[\/latex] and [latex]f(x)=(x-k)q(x)+0[\/latex] or [latex]f(x)=(x-k)q(x).[\/latex]<\/p>\n<p id=\"fs-id1165135176357\">Notice, written in this form, [latex]x-k[\/latex] is a factor of [latex]f(x).[\/latex] We can conclude if [latex]k[\/latex] is a zero of [latex]f(x),[\/latex] then [latex]x-k[\/latex] is a factor of [latex]f(x).[\/latex]<\/p>\n<p id=\"fs-id1165135684373\">Similarly, if [latex]x-k[\/latex] is a factor of [latex]f(x),[\/latex] then the remainder of the Division Algorithm [latex]f(x)=(x-k)q(x)+r[\/latex] is 0. This tells us that [latex]k[\/latex] is a zero.<\/p>\n<p id=\"fs-id1165132943504\">This pair of implications is the Factor Theorem. As we will soon see, a polynomial of degree [latex]n[\/latex] in the complex number system will have [latex]n[\/latex] zeros. We can use the Factor Theorem to completely factor a polynomial into the product of [latex]n[\/latex] factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">The Factor Theorem<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>According to the Factor Theorem, [latex]k[\/latex] is a zero of [latex]f(x)[\/latex] if and only if [latex](x-k)[\/latex] is a factor of [latex]f(x).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a factor and a third-degree polynomial, use the Factor Theorem to factor the polynomial.<\/strong><\/p>\n<ol>\n<li>Use synthetic division to divide the polynomial by [latex](x-k).[\/latex]<\/li>\n<li>Confirm that the remainder is 0.<\/li>\n<li>Write the polynomial as the product of [latex](x-k)[\/latex] and the quadratic quotient.<\/li>\n<li>If possible, factor the quadratic.<\/li>\n<li>Write the polynomial as the product of factors.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div><\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Using the Factor Theorem to Find the Zeros of a Polynomial Expression<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Show that [latex](x+2)[\/latex] is a factor of [latex]x^3-6x^2-x+30.[\/latex] Find the remaining factors. Use the factors to determine the zeros of the polynomial.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We can use synthetic division to show that [latex](x+2)[\/latex] is a factor of the polynomial.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-854 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-2-300x71.jpeg\" alt=\"\" width=\"300\" height=\"71\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-2-300x71.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-2-65x15.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-2-225x53.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-2.jpeg 325w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>The remainder is zero, so [latex](x+2)[\/latex] is a factor of the polynomial. We can use the Division Algorithm to write the polynomial as the product of the divisor and the quotient:<\/p>\n<p style=\"text-align: center;\">[latex](x+2)(x^2-8x+15)[\/latex]<\/p>\n<p>We can factor the quadratic factor to write the polynomial as<\/p>\n<p style=\"text-align: center;\">[latex](x+2)(x-3)(x-5)[\/latex]<\/p>\n<p>By the Factor Theorem, the zeros of [latex]x^3-6x^2-x+30[\/latex] are [latex]-2, 3 \\ \\ \\text{and} \\ \\ 5.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div><\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use the Factor Theorem to find the zeros of [latex]f(x)=x^3+4x^2-4x-16[\/latex] given that [latex](x-2)[\/latex] is a factor of the polynomial.<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165134152972\" data-depth=\"1\">\n<h2 data-type=\"title\">Using the Rational Zero Theorem to Find Rational Zeros<\/h2>\n<p id=\"fs-id1165137660817\">Another use for the Remainder Theorem is to test whether a rational number is a zero for a given polynomial. But first we need a pool of rational numbers to test. The <strong><span id=\"term-00010\" data-type=\"term\">Rational Zero Theorem<\/span><\/strong> helps us to narrow down the number of possible rational zeros using the ratio of the factors of the constant term and factors of the leading <span id=\"term-00011\" class=\"no-emphasis\" data-type=\"term\">coefficient<\/span> of the polynomial<\/p>\n<p id=\"fs-id1165135508309\">Consider a quadratic function with two zeros, [latex]x=\\frac{2}{5}[\/latex] and [latex]x=\\frac{3}{4}.[\/latex] By the Factor Theorem, these zeros have factors associated with them. Let us set each factor equal to 0, and then construct the original quadratic function absent its stretching factor.<\/p>\n<p><span id=\"eip-id1165135315549\" data-type=\"media\" data-alt=\"\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-855 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-rational-zero-300x65.jpeg\" alt=\"\" width=\"420\" height=\"91\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-rational-zero-300x65.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-rational-zero-768x167.jpeg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-rational-zero-65x14.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-rational-zero-225x49.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-rational-zero-350x76.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-rational-zero.jpeg 781w\" sizes=\"auto, (max-width: 420px) 100vw, 420px\" \/><\/span><\/p>\n<p id=\"fs-id1165135485170\">Notice that two of the factors of the constant term, 6, are the two numerators from the original rational roots: 2 and 3. Similarly, two of the factors from the leading coefficient, 20, are the two denominators from the original rational roots: 5 and 4.<\/p>\n<p id=\"fs-id1165137761317\">We can infer that the numerators of the rational roots will always be factors of the constant term and the denominators will be factors of the leading coefficient. This is the essence of the Rational Zero Theorem; it is a means to give us a pool of possible rational zeros.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">The Rational Zero Theorem<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The <span id=\"term-00012\" data-type=\"term\">Rational Zero Theorem<\/span> states that, if the polynomial [latex]f(x)=a_nx^n+a_{n-1}x^{n-1}+\\ldots +a_1x+a_0[\/latex] has integer coefficients [latex]a_n\\not=0,[\/latex] then every rational zero of [latex]f(x)[\/latex] has the form [latex]\\frac{p}{q}[\/latex] where [latex]p[\/latex] is a factor of the constant term [latex]a_0[\/latex] and [latex]q[\/latex] is a factor of the leading coefficient [latex]a_n.[\/latex]<\/p>\n<p>When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a polynomial function [latex]f(x)[\/latex] use the Rational Zero Theorem to find rational zeros.<\/strong><\/p>\n<ol>\n<li>Determine all factors of the constant term and all factors of the leading coefficient.<\/li>\n<li>Determine all possible values of [latex]\\frac{p}{q},[\/latex] where [latex]p[\/latex] is a factor of the constant term and [latex]q[\/latex] is a factor of the leading coefficient. Be sure to include both positive and negative candidates.<\/li>\n<li>Determine which possible zeros are actual zeros by evaluating each case of [latex]f(\\frac{p}{q}).[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div><\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Listing All Possible Rational Zeros<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1165135640968\">List all possible rational zeros of [latex]f(x)=2x^4-5x^3+x^2-4.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The only possible rational zeros of [latex]f(x)[\/latex] are the quotients of the factors of the last term, [latex]-4,[\/latex] and the factors of the leading coefficient, 2.<\/p>\n<p>The constant term is [latex]-4;[\/latex] the factors of [latex]-4[\/latex] are [latex]p=\\pm1, \\pm2, \\pm4.[\/latex]<\/p>\n<p>The leading coefficient is 2; the factors of 2 are [latex]q=\\pm1, \\pm2.[\/latex]<\/p>\n<p>If any of the four real zeros are rational zeros, then they will be of one of the following factors of [latex]-4[\/latex] divided by one of the factors of 2.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll} \\frac{p}{q}=\\pm\\frac{1}{1}, \\pm\\frac{1}{2} && \\frac{p}{q}=\\pm\\frac{2}{1}, \\pm\\frac{2}{2} && \\frac{p}{q}=\\pm\\frac{4}{1}, \\pm\\frac{4}{2} \\end{array}[\/latex]<\/p>\n<p>Note that [latex]\\frac{2}{2}=1[\/latex] and [latex]\\frac{4}{2}=2,[\/latex] which have already been listed. So we can shorten our list<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{p}{q}=\\frac{\\text{Factors of the last}}{\\text{Factors of the first}}=\\pm1, \\pm2, \\pm4, \\pm\\frac{1}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Using the Rational Zero Theorem to Find Rational Zeros<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use the Rational Zero Theorem to find the rational zeros of [latex]f(x)=2x^3+x^2-4x+1.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The Rational Zero Theorem tells us that if [latex]\\frac{p}{q}[\/latex] is a zero of [latex]f(x),[\/latex] then [latex]p[\/latex] is a factor of 1 and [latex]q[\/latex] is a factor of 2.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} \\frac{p}{q} &=& \\frac{\\text{factor of constant term}}{\\text{factor of leading coefficient}} \\\\ &=& \\frac{\\text{factor of } 1}{\\text{factor of } 2} \\end{array}[\/latex]<\/p>\n<p>The factors of 1 are [latex]\\pm1[\/latex] and the factors of 2 are [latex]\\pm1[\/latex] and [latex]\\pm2.[\/latex] The possible values for [latex]\\frac{p}{q}[\/latex] are [latex]\\pm1[\/latex] and [latex]\\pm\\frac{1}{2}.[\/latex] These are the possible rational zeros for the function. We can determine which of the possible zeros are actual zeros by substituting these values for [latex]x[\/latex] in [latex]f(x).[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} f(-1) &=& 2(-1)^3+(-1)^2-4(-1)+1 &=& 4 \\\\ f(1) &=& 2(1)^3+(1)^2-4(1)+1 &=& 0 \\\\ f(-\\frac{1}{2}) &=& 2(-\\frac{1}{2})^3+(-\\frac{1}{2})^2-4(-\\frac{1}{2})+1 &=& 3 \\\\ f(\\frac{1}{2}) &=& 2(\\frac{1}{2})^3+(\\frac{1}{2})^2-4(\\frac{1}{2})+1 &=& -\\frac{1}{2} \\end{array}[\/latex]<\/p>\n<p>Of those, [latex]-1, -\\frac{1}{2}, \\ \\text{and } \\frac{1}{2}[\/latex] are not zeros of [latex]f(x).[\/latex] 1 is the only rational zero of [latex]f(x).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use the Rational Zero Theorem to find the rational zeros of [latex]f(x)=x^3-5x^2+2x+1.[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165137724990\" data-depth=\"1\">\n<h2 data-type=\"title\">Finding the Zeros of Polynomial Functions<\/h2>\n<p id=\"fs-id1165135530405\">The Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. Once we have done this, we can use synthetic division repeatedly to determine all of the <strong>zeros<\/strong> of a polynomial function.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a polynomial function <\/strong> [latex]f,[\/latex] use synthetic division to find its zeros.<\/p>\n<ol>\n<li>Use the Rational Zero Theorem to list all possible rational zeros of the function.<\/li>\n<li>Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the polynomial. If the remainder is 0, the candidate is a zero. If the remainder is not zero, discard the candidate.<\/li>\n<li>Repeat step two using the quotient found with synthetic division. If possible, continue until the quotient is a quadratic.<\/li>\n<li>Find the zeros of the quadratic function. Two possible methods for solving quadratics are factoring and using the quadratic formula.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 5: Finding the Zeros of a Polynomial Function with Repeated Real Zeros<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the zeros of [latex]f(x)=4x^3-3x-1.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The Rational Zero Theorem tells us that if [latex]\\frac{p}{q}[\/latex] is a zero of [latex]f(x),[\/latex] then [latex]p[\/latex] is a factor of -1 and [latex]q[\/latex] is a factor of 4.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} \\frac{p}{q} &=& \\frac{\\text{factor of constant term}}{\\text{factor of leading coefficient}} \\\\ &=& \\frac{\\text{factor of } -1}{\\text{factor of } 4} \\end{array}[\/latex]<\/p>\n<p>The factors of [latex]-1[\/latex] are [latex]\\pm1[\/latex] and the factors of [latex]4[\/latex] are [latex]\\pm1, \\pm2,[\/latex] and [latex]\\pm4.[\/latex] The possible values for [latex]\\frac{p}{q}[\/latex] are [latex]\\pm1, \\pm\\frac{1}{2},[\/latex] and [latex]\\pm\\frac{1}{4}.[\/latex] These are the possible rational zeros for the function. We will use synthetic division to evaluate each possible zero until we find one that gives a remainder of 0. Let\u2019s begin with 1.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-856 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-5-300x71.jpeg\" alt=\"\" width=\"300\" height=\"71\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-5-300x71.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-5-65x15.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-5-225x53.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-5.jpeg 325w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Dividing by [latex](x-1)[\/latex] gives a remainder of 0, so 1 is a zero of the function. The polynomial can be written as<\/p>\n<p style=\"text-align: center;\">[latex](x-1)(4x^2+4x+1)[\/latex]<\/p>\n<p>The quadratic is a perfect square. [latex]f(x)[\/latex] can be written as<\/p>\n<p style=\"text-align: center;\">[latex](x-1)(2x+1)^2[\/latex]<\/p>\n<p>We already know that 1 is a zero. The other zero will have a multiplicity of 2 because the factor is squared. To find the other zero, we can set the factor equal to 0.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rr} 2x+1 &=& 0 \\\\ x &=& -\\frac{1}{2} \\end{array}[\/latex]<\/p>\n<p>The zeros of the function are 1 and [latex]-\\frac{1}{2}[\/latex] with multiplicity 2.<\/p>\n<h3>Analysis<\/h3>\n<p>Look at the graph of the function [latex]f[\/latex] in Figure 2. Notice, at [latex]x=-0.5,[\/latex] the graph bounces off the <em data-effect=\"italics\">x<\/em>-axis, indicating the even multiplicity (2,4,6\u2026) for the zero [latex]-0.5.[\/latex] At [latex]x=1,[\/latex] the graph crosses the <em data-effect=\"italics\">x<\/em>-axis, indicating the odd multiplicity (1,3,5\u2026) for the zero [latex]x=1.[\/latex]<\/p>\n<figure id=\"attachment_857\" aria-describedby=\"caption-attachment-857\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-857\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-1-300x178.jpeg\" alt=\"\" width=\"300\" height=\"178\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-1-300x178.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-1-65x39.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-1-225x134.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-1-350x208.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-1.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-857\" class=\"wp-caption-text\">Figure 2<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<section id=\"fs-id1165137461780\" data-depth=\"1\">\n<h2 data-type=\"title\">Using the Fundamental Theorem of Algebra<\/h2>\n<p id=\"fs-id1165135547255\">Now that we can find rational zeros for a polynomial function, we will look at a theorem that discusses the number of complex zeros of a polynomial function. The <strong>Fundamental Theorem of Algebra <\/strong>tells us that every polynomial function has at least one complex zero. This theorem forms the foundation for solving polynomial equations.<\/p>\n<p id=\"fs-id1165137771429\">Suppose [latex]f[\/latex] is a polynomial function of degree four, and [latex]f(x)=0.[\/latex] The Fundamental Theorem of Algebra states that there is at least one complex solution, call it [latex]c_1.[\/latex] By the Factor Theorem, we can write [latex]f(x)[\/latex] as a product of [latex]x-c_1[\/latex] and a polynomial quotient. Since [latex]x-c_1[\/latex] is linear, the polynomial quotient will be of degree three. Now we apply the Fundamental Theorem of Algebra to the third-degree polynomial quotient. It will have at least one complex zero, call it [latex]c_2.[\/latex] So we can write the polynomial quotient as a product of [latex]x-c_2[\/latex] and a new polynomial quotient of degree two. Continue to apply the Fundamental Theorem of Algebra until all of the zeros are found. There will be four of them and each one will yield a factor of [latex]f(x).[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">The Fundamental Theorem of Algebra<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The <strong><span id=\"term-00015\" data-type=\"term\">Fundamental Theorem of Algebra<\/span><\/strong> states that, if [latex]f(x)[\/latex] is a polynomial of degree [latex]n> 0,[\/latex] then [latex]f(x)[\/latex] has at least one complex zero.<\/p>\n<p>We can use this theorem to argue that, if [latex]f(x)[\/latex] is a polynomial of degree [latex]n> 0,[\/latex] and [latex]a[\/latex] is a non-zero real number, then [latex]f(x)[\/latex] has exactly [latex]n[\/latex] linear factors<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=a(x-c_1)(x-c_2)\\ldots(x-c_n)[\/latex]<\/p>\n<p>where [latex]c_1, c_2, \\ldots, c_n[\/latex] are complex numbers. Therefore, [latex]f(x)[\/latex] has [latex]n[\/latex] roots if we allow for multiplicities.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<div class=\"os-note-body\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Q: Does every polynomial have at least one imaginary zero?<\/strong><\/p>\n<p>A: <em data-effect=\"italics\">No. Real numbers are a subset of complex numbers, but not the other way around. A complex number is not necessarily imaginary. Real numbers are also complex numbers.<\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 6: Finding the Zeros of a Polynomial Function with Complex Zeros<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the zeros of [latex]f(x)=3x^3+9x^2+x+3.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The Rational Zero Theorem tells us that if [latex]\\frac{p}{q}[\/latex] is a zero of [latex]f(x),[\/latex] then [latex]p[\/latex] is a factor of 3 and [latex]q[\/latex] is a factor of 3.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} \\frac{p}{q} &=& \\frac{\\text{factor of constant term}}{\\text{factor of leading coefficient}} \\\\ &=& \\frac{\\text{factor of } 3}{\\text{factor of } 3} \\end{array}[\/latex]<\/p>\n<p>The factors of 3 are [latex]\\pm1[\/latex] and [latex]\\pm3.[\/latex] The possible values for [latex]\\frac{p}{q}[\/latex] and therefore the possible rational zeros for the function, are [latex]\\pm3, \\pm1, \\ \\text{and } \\pm\\frac{1}{3}.[\/latex] We will use synthetic division to evaluate each possible zero until we find one that gives a remainder of 0. Let\u2019s begin with \u20133.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-858 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-6-300x71.jpeg\" alt=\"\" width=\"300\" height=\"71\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-6-300x71.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-6-65x15.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-6-225x53.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-6.jpeg 325w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Dividing by [latex](x+3)[\/latex] gives a remainder of 0, so [latex]-3[\/latex] is a zero of the function. The polynomial can be written as<\/p>\n<p style=\"text-align: center;\">[latex](x+3)(3x^2+1)[\/latex]<\/p>\n<p>We can then set the quadratic equal to 0 and solve to find the other zeros of the function.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rr} 3x^2+1 &=& 0 \\\\ x^2 &=& -\\frac{1}{3} \\\\ x &=& \\pm\\sqrt{-\\frac{1}{3}} &=& \\pm\\frac{i\\sqrt{3}}{3} \\end{array}[\/latex]<\/p>\n<p>The zeros [latex]f(x)[\/latex] of are [latex]-3[\/latex] and [latex]\\pm\\frac{i\\sqrt{3}}{3}.[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>Look at the graph of the function [latex]f[\/latex] in Figure 3. Notice that, at [latex]x=-3,[\/latex] the graph crosses the <em data-effect=\"italics\">x<\/em>-axis, indicating an odd multiplicity (1) for the zero [latex]x=-3.[\/latex] Also note the presence of the two turning points. This means that, since there is a 3<sup>rd<\/sup> degree polynomial, we are looking at the maximum number of turning points. So, the end behavior of increasing without bound to the right and decreasing without bound to the left will continue. Thus, all the <em data-effect=\"italics\">x<\/em>-intercepts for the function are shown. So either the multiplicity of [latex]x=-3[\/latex] is 1 and there are two complex solutions, which is what we found, or the multiplicity at [latex]x=-3[\/latex] is three. Either way, our result is correct.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_859\" aria-describedby=\"caption-attachment-859\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-859\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-2-300x178.jpeg\" alt=\"\" width=\"300\" height=\"178\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-2-300x178.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-2-65x39.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-2-225x134.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-2-350x208.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-2.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-859\" class=\"wp-caption-text\">Figure 3<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<section><\/section>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the zeros of [latex]f(x)=2x^3+5x^2-11x+4.[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165135501998\" data-depth=\"1\">\n<h2 data-type=\"title\">Using the Linear Factorization Theorem to Find Polynomials with Given Zeros<\/h2>\n<p id=\"fs-id1165135502003\">A vital implication of the <span id=\"term-00016\" class=\"no-emphasis\" data-type=\"term\">Fundamental Theorem of Algebra<\/span>, as we stated above, is that a polynomial function of degree [latex]n[\/latex] will have [latex]n[\/latex] zeros in the set of complex numbers, if we allow for multiplicities. This means that we can factor the polynomial function into [latex]n[\/latex] factors. The<strong> Linear Factorization Theorem<\/strong> tells us that a polynomial function will have the same number of factors as its degree, and that each factor will be in the form [latex](x-c),[\/latex] where [latex]c[\/latex] is a complex number.<\/p>\n<p id=\"eip-651\">Let [latex]f[\/latex] be a polynomial function with real coefficients, and suppose [latex]a+bi, b\\not=0,[\/latex] is a zero of [latex]f(x).[\/latex] Then, by the Factor Theorem, [latex]x-(a+bi)[\/latex] is a factor of [latex]f(x).[\/latex] For [latex]f[\/latex] to have real coefficients, [latex]x-(a-bi)[\/latex] must also be a factor of [latex]f(x).[\/latex] This is true because any factor other than [latex]x-(a-bi)[\/latex] when multiplied by [latex]x-(a+bi),[\/latex] will leave imaginary components in the product. Only multiplication with conjugate pairs will eliminate the imaginary parts and result in real coefficients. In other words, if a polynomial function [latex]f[\/latex] with real coefficients has a complex zero [latex]a+bi,[\/latex] then the complex conjugate [latex]a-bi[\/latex] must also be a zero of [latex]f(x).[\/latex] This is called the Complex Conjugate Theorem.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Complex Conjugate Theorem<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>According to the <strong>Linear Factorization Theorem<\/strong>, a polynomial function will have the same number of factors as its degree, and each factor will be in the form [latex](x-c),[\/latex] where [latex]c[\/latex] is a complex number.<\/p>\n<p>If the polynomial function [latex]f[\/latex] has real coefficients and a complex zero in the form [latex]a+bi,[\/latex] then the complex conjugate of the zero, [latex]a-bi,[\/latex] is also a zero.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given the zeros of a polynomial function [latex]f[\/latex] and a point [latex](c, f(c))[\/latex] on the graph of [latex]f,[\/latex] use the Linear Factorization Theorem to find the polynomial function.<\/strong><\/p>\n<ol>\n<li>Use the zeros to construct the linear factors of the polynomial.<\/li>\n<li>Multiply the linear factors to expand the polynomial.<\/li>\n<li>Substitute [latex](c, f(c))[\/latex] into the function to determine the leading coefficient<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 7: Using the Linear Factorization Theorem to Find a Polynomial with Given Zeros<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find a fourth degree polynomial with real coefficients that has zeros of [latex]-3, 2, i,[\/latex] such that [latex]f(-2)=100.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Because [latex]x=i[\/latex] is a zero, by the Complex Conjugate Theorem [latex]x=-i[\/latex] is also a zero. The polynomial must have factors of [latex](x+3), (x-2), (x-i),[\/latex] and [latex](x+i).[\/latex] Since we are looking for a degree 4 polynomial, and now have four zeros, we have all four factors. Let\u2019s begin by multiplying these factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} f(x) &=& a(x+3)(x-2)(x-i)(x+i) \\\\ f(x) &=& a(x^2+x-6)(x^2+1) \\\\ f(x) &=& a(x^4+x^3-5x^2+x-6) \\end{array}[\/latex]<\/p>\n<p>We need to find [latex]a[\/latex] to ensure [latex]f(-2)=100.[\/latex] Substitute [latex]x=-2[\/latex] and [latex]f(2)=100[\/latex] into [latex]f(x).[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} 100 &=& a((-2)^4+(-2)^3-5(-2)^2+(-2)-6) \\\\ 100 &=& a(-2) \\\\ -5 &=& a \\end{array}[\/latex]<\/p>\n<p>So the polynomial function is<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=-5(x^4+x^3-5x^2+x-6)[\/latex]<\/p>\n<p>or<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=-5x^4-5x^3+25x^2-5x+30[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>We found that both [latex]i[\/latex] and [latex]-i[\/latex] were zeros, but only one of these zeros needed to be given. If [latex]i[\/latex] is a zero of a polynomial with real coefficients, then [latex]-i[\/latex] must also be a zero of the polynomial because is the complex conjugate of [latex]i.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<section><\/section>\n<\/div>\n<div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Q: If <\/strong> [latex]2+3i[\/latex] were given as a zero of a polynomial with real coefficients, would [latex]2-3i[\/latex] also need to be a zero?<\/p>\n<p>A: <em data-effect=\"italics\">Yes. When any complex number with an imaginary component is given as a zero of a polynomial with real coefficients, the conjugate must also be a zero of the polynomial.<\/em><\/p>\n<\/div>\n<\/div>\n<section><\/section>\n<\/div>\n<div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find a third degree polynomial with real coefficients that has zeros of 5 and [latex]-2i[\/latex] such that [latex]f(1)=10.[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165135177650\" data-depth=\"1\">\n<h2 data-type=\"title\">Using Descartes\u2019 Rule of Signs<\/h2>\n<p id=\"fs-id1165135177655\">There is a straightforward way to determine the possible numbers of positive and negative real zeros for any polynomial function. If the polynomial is written in descending order,<strong> Descartes\u2019 Rule of Signs<\/strong> tells us of a relationship between the number of sign changes in [latex]f(x)[\/latex] and the number of positive real zeros. For example, the polynomial function below has one sign change.<span id=\"fs-id1165134378690\" data-type=\"media\" data-alt=\"The function, f(x)=x^4+x^3+x^2+x-1, has one sign change between x and -1.&#96;\" data-display=\"block\"><\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1560 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-300x28.webp\" alt=\"\" width=\"321\" height=\"30\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-300x28.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-65x6.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-225x21.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-350x32.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5.webp 487w\" sizes=\"auto, (max-width: 321px) 100vw, 321px\" \/><\/p>\n<p id=\"fs-id1165135206081\">This tells us that the function must have 1 positive real zero.<\/p>\n<p id=\"fs-id1165135206084\">There is a similar relationship between the number of sign changes in [latex]f(-x)[\/latex] and the number of negative real zeros.<span id=\"fs-id1165135152070\" data-type=\"media\" data-alt=\"The function, f(-x)=(-x)^4+(-x)^3+(-x)^2+(-x)-1=+ x^4-x^3+x^2-x-1, has three sign changes between x^4 and x^3, x^3 and x^2, and x^2 and x.&#96;\" data-display=\"block\"><\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1561 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5.1-300x47.webp\" alt=\"\" width=\"345\" height=\"54\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5.1-300x47.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5.1-65x10.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5.1-225x35.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5.1-350x55.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5.1.webp 487w\" sizes=\"auto, (max-width: 345px) 100vw, 345px\" \/><\/p>\n<p id=\"fs-id1165135152083\">In this case, [latex]f(-x)[\/latex] has 3 sign changes. This tells us that [latex]f(x)[\/latex] could have 3 or 1 negative real zeros.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Descartes&#8217; Rule of Signs<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>According to <strong>Descartes\u2019 Rule of Signs<\/strong>, if we let [latex]f(x)=a_nx^n+a_{n-1}x^{n-1}+\\ldots+a_1x+a_0[\/latex] be a polynomial function with real coefficients:<\/p>\n<ul>\n<li>The number of positive real zeros is either equal to the number of sign changes of [latex]f(x)[\/latex] or is less than the number of sign changes by an even integer.<\/li>\n<li>The number of negative real zeros is either equal to the number of sign changes of [latex]f(-x)[\/latex] or is less than the number of sign changes by an even integer.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 8: Using Descartes&#8217; Rule of Signs<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use Descartes\u2019 Rule of Signs to determine the possible numbers of positive and negative real zeros for [latex]f(x)=-x^4-3x^3+6x^2-4x-12.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Begin by determining the number of sign changes.<\/p>\n<figure id=\"attachment_860\" aria-describedby=\"caption-attachment-860\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-860\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-3-300x32.jpeg\" alt=\"\" width=\"300\" height=\"32\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-3-300x32.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-3-65x7.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-3-225x24.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-3-350x37.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-3.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-860\" class=\"wp-caption-text\">Figure 4<\/figcaption><\/figure>\n<p>There are two sign changes, so there are either 2 or 0 positive real roots. Next, we examine [latex]f(-x)[\/latex] to determine the number of negative real roots.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} f(-x) &=& -(-x)^4--3(-x)^3+6(-x)^2-4(-x)-12 \\\\ f(-x) &=& -x^4+3x^3+6x^2+4x-12 \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_861\" aria-describedby=\"caption-attachment-861\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-861\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-4-300x30.jpeg\" alt=\"\" width=\"300\" height=\"30\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-4-300x30.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-4-65x6.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-4-225x22.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-4-350x34.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-4.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-861\" class=\"wp-caption-text\">Figure 5<\/figcaption><\/figure>\n<p>Again, there are two sign changes, so there are either 2 or 0 negative real roots.<\/p>\n<p>There are four possibilities, as we can see in Table 1.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 1<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 25%; text-align: center;\"><strong>Positive Real Zeros<\/strong><\/td>\n<td style=\"width: 25%; text-align: center;\"><strong>Negative Real Zeros<\/strong><\/td>\n<td style=\"width: 25%; text-align: center;\"><strong>Complex Zeros<\/strong><\/td>\n<td style=\"width: 25%; text-align: center;\"><strong>Total Zeros<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%; text-align: center;\">2<\/td>\n<td style=\"width: 25%; text-align: center;\">2<\/td>\n<td style=\"width: 25%; text-align: center;\">0<\/td>\n<td style=\"width: 25%; text-align: center;\">4<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%; text-align: center;\">2<\/td>\n<td style=\"width: 25%; text-align: center;\">0<\/td>\n<td style=\"width: 25%; text-align: center;\">2<\/td>\n<td style=\"width: 25%; text-align: center;\">4<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%; text-align: center;\">0<\/td>\n<td style=\"width: 25%; text-align: center;\">2<\/td>\n<td style=\"width: 25%; text-align: center;\">2<\/td>\n<td style=\"width: 25%; text-align: center;\">4<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%; text-align: center;\">0<\/td>\n<td style=\"width: 25%; text-align: center;\">0<\/td>\n<td style=\"width: 25%; text-align: center;\">4<\/td>\n<td style=\"width: 25%; text-align: center;\">4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Analysis<\/h3>\n<p>We can confirm the numbers of positive and negative real roots by examining a graph of the function. See Figure 6. We can see from the graph that the function has 0 positive real roots and 2 negative real roots.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_862\" aria-describedby=\"caption-attachment-862\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-862\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-5-300x153.jpeg\" alt=\"\" width=\"300\" height=\"153\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-5-300x153.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-5-65x33.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-5-225x115.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-5-350x179.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-fig-5.jpeg 596w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-862\" class=\"wp-caption-text\">Figure 6<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<section><\/section>\n<\/div>\n<div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use Descartes\u2019 Rule of Signs to determine the maximum possible numbers of positive and negative real zeros for [latex]f(x)=2x^4-10x^3+11x^2-15x+12.[\/latex] Use a graph to verify the numbers of positive and negative real zeros for the function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<section id=\"fs-id1165135440213\" data-depth=\"1\">\n<h2 data-type=\"title\">Solving Real-World Applications<\/h2>\n<p id=\"fs-id1165135440219\">We have now introduced a variety of tools for solving polynomial equations. Let\u2019s use these tools to solve the bakery problem from the beginning of the section.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 9: Solving Polynomial Equations<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A new bakery offers decorated, multi-tiered cakes for display and cutting at Quincea\u00f1era and wedding celebrations, as well as sheet cakes to serve most of the guests. The bakery wants the volume of a small sheet cake to be 351 cubic inches. The cake is in the shape of a rectangular solid. They want the length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of the width. What should the dimensions of the cake pan be?<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Begin by writing an equation for the volume of the cake. The volume of a rectangular solid is given by [latex]V=lwh.[\/latex] We were given that the length must be four inches longer than the width, so we can express the length of the cake as [latex]l=w+4.[\/latex] We were given that the height of the cake is one-third of the width, so we can express the height of the cake as [latex]h=\\frac{1}{3}w.[\/latex] Let\u2019s write the volume of the cake in terms of width of the cake.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} V &=& (w+4)(w)(\\frac{1}{3}w) \\\\ V &=& \\frac{1}{3}w^3+\\frac{4}{3}w^2 \\end{array}[\/latex]<\/p>\n<p>Substitute the given volume into this equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} 351 &=& \\frac{1}{3}w^3+\\frac{4}{3}w^2 & \\quad \\text{Substitue 351 for } V. \\\\ 1053 &=& w^3+4w^2 & \\quad \\text{Multiply both sides by 3.} \\\\ 0 &=& w^3+4w^2-1053 & \\quad \\text{Subtract 1-53 from both sides.} \\end{array}[\/latex]<\/p>\n<p>Descartes&#8217; rule of signs tells us there is one positive solution. The Rational Zero Theorem tells us that the possible rational zeros are [latex]\\pm1, \\pm3, \\pm9, \\pm13, \\pm27, \\pm39, \\pm81, \\pm117, \\pm351[\/latex] and [latex]\\pm1053.[\/latex] We can use synthetic division to test these possible zeros. Only positive numbers make sense as dimensions for a cake, so we need not test any negative values. Let\u2019s begin by testing values that make the most sense as dimensions for a small sheet cake. Use synthetic division to check [latex]x=1.[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-863 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-9.1-300x71.jpeg\" alt=\"\" width=\"300\" height=\"71\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-9.1-300x71.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-9.1-65x15.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-9.1-225x53.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-9.1.jpeg 325w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Since 1 is not a solution, we will check [latex]x=3.[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-864 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-9.2-300x71.jpeg\" alt=\"\" width=\"300\" height=\"71\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-9.2-300x71.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-9.2-65x15.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-9.2-225x53.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-9.2.jpeg 325w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Since 3 is not a solution either, we will test [latex]x=9.[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-865 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-9.3-300x71.jpeg\" alt=\"\" width=\"300\" height=\"71\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-9.3-300x71.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-9.3-65x15.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-9.3-225x53.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.5-ex-9.3.jpeg 325w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Synthetic division gives a remainder of 0, so 9 is a solution to the equation. We can use the relationships between the width and the other dimensions to determine the length and height of the sheet cake pan.<\/p>\n<p style=\"text-align: center;\">[latex]l=w+4=9+4=13 \\ \\text{and } h=\\frac{1}{3}w=\\frac{1}{3}(9)=3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A shipping container in the shape of a rectangular solid must have a volume of 84 cubic meters. The client tells the manufacturer that, because of the contents, the length of the container must be one meter longer than the width, and the height must be one meter greater than twice the width. What should the dimensions of the container be?<\/p>\n<\/div>\n<\/div>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_03_06_07\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137843250\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Media<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Access these online resources for additional instruction and practice with zeros of polynomial functions.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=e_EttLeQblY\">Real Zeros, Factors, and Graphs of Polynomial Functions<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=F0bcRJfdh5Q\">Complex Factorization Theorem<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=LeZdCSCIb3Q\">Find the Zeros of a Polynomial Function<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=D_I11k2DfCg\">Find the Zeros of a Polynomial Function 2<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=paq5VWwXHp8\">Find the Zeros of a Polynomial Function 3<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">5.5 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1165135149755\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1165135149759\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1165135149765\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135149766\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135149765-solution\">1<\/a><span class=\"os-divider\">. <\/span>Describe a use for the Remainder Theorem.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135149774\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137898784\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>Explain why the Rational Zero Theorem does not guarantee finding zeros of a polynomial function.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137898789\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137898790\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137898789-solution\">3<\/a><span class=\"os-divider\">. <\/span>What is the difference between rational and real zeros?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137898800\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137898801\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>If Descartes\u2019 Rule of Signs reveals a no change of signs or one sign of changes, what specific conclusion can be drawn?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137898807\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137898808\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137898807-solution\">5<\/a><span class=\"os-divider\">. <\/span>If synthetic division reveals a zero, why should we try that value again as a possible solution?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135531376\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id1165135531382\">For the following exercises, use the Remainder Theorem to find the remainder.<\/p>\n<div id=\"fs-id1165135531385\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135531386\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex](x^4-9x^2+14)\\div (x-2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134152708\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134152709\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134152708-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex](3x^3=2x^2+x-4)\\div (x+3)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135532325\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135532326\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex](x^4+5x^3-4x-17)\\div (x+1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135571875\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135571876\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135571875-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex](-3x^2+6x+24)\\div (x-4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134106023\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137675202\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex](5x^5-4x^4+3x^3-2x^2+x-1)\\div (x+6)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135487281\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135487282\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135487281-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex](x^4-1)\\div (x-4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137933131\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137933132\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex](3x^3+4x^2-8x+2)\\div (x-3)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134430485\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134430486\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134430485-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex](4x^3+5x^2-2x+7)\\div (x+2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135403423\">For the following exercises, use the given factor and the Factor Theorem to find all real zeros for the given polynomial function.<\/p>\n<div id=\"fs-id1165135403427\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135403428\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=2x^3-9x^2+13x-6; x-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135349107\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135349108\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135349107-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=2x^3+x^2-5x+2; x+2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135198461\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135198462\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=3x^3+x^2-20x+12; x+3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137646943\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137646944\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137646943-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=2x^3+3x^2+x+6; x+2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135363964\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135363965\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=-5x^3+16x^2-9; x-3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137833870\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137833871\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137833870-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=x^3+3x^2+4x+12; x+3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134039305\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134039306\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=4x^3-7x+3; x-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135453082\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135453083\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135453082-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=2x^3+5x^2-12x-30; 2x+5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165134472281\">For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.<\/p>\n<div id=\"fs-id1165134472284\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134472285\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex]x^3-3x^2-10x+24=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134212161\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134212162\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134212161-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex]2x^3+7x^2-10x-24=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135678616\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135678617\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex]x^3+2x^2-9x-18=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135554358\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135554359\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135554358-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex]x^3+5x^2-16x-80=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135154364\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135154365\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex]x^3-3x^2-25x+75=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135389007\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133093326\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135389007-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex]2x^3-3x^2-32x-15=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137676884\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137676885\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex]2x^3+x^2-7x-6=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135393418\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135393419\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135393418-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex]2x^3-3x^2-x+1=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137766744\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137766746\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex]3x^3-x^2-11x-6=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135356524\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135632075\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135356524-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex]2x^3-5x^2+9x-9=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135695187\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135695188\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex]2x^3-3x^2+4x+3=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135692865\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135692866\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135692865-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex]x^4-2x^3-7x^2+8x+12=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134474183\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134474184\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex]x^4+2x^3-9x^2-2x+8=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135443739\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135443740\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135443739-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex]4x^4+4x^3-25x^2-x+6=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137680607\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137680608\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex]2x^4-3x^3-15x^2+32x-12=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135353111\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135353112\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135353111-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex]x^4+2x^3-4x^2-10x-5=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133311066\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133311068\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex]4x^3-3x+1=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133214917\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133214918\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165133214917-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex]8x^4+26x^3+39x^2+26x+6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135517156\">For the following exercises, find all complex solutions (real and non-real).<\/p>\n<div id=\"fs-id1165135517160\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135517161\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex]x^3+x62+x+1=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132940010\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132940011\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165132940010-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex]x^3-8x^2+25x-26=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135199514\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135199515\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex]x^3+13x^2+57x+85=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134380350\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134380351\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134380350-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex]3x^3-4x^2+11x+10=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135416512\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135416513\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex]x^4+2x^3+22x^2+50x-75=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137851467\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137851468\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137851467-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex]2x^3-3x^2+32x+17=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165135403510\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphical<\/h3>\n<p id=\"fs-id1165135403515\">For the following exercises, use Descartes\u2019 Rule to determine the possible number of positive and negative solutions. Confirm with the given graph.<\/p>\n<div id=\"fs-id1165135403521\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135403522\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=x^3-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165134061970\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134061972\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134061970-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=x^4-x^2-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135333211\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135333212\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=x^3-2x^2-5x+6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135485740\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135485741\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135485740-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=x^3-2x^2+x-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134261871\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134261872\" data-type=\"problem\">\n<p><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=x^4+2x^3-12x^2+14x-5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134149873\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134149874\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134149873-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=2x^3+37x^2+200x+300[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134389900\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134389901\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=x^3-2x^2-16x+32[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134231463\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134231464\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134231463-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=2x^4-5x^3-5x^2+5x+3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134148467\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134148468\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=2x^4-5x^3-14x^2+20x+8[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135640945\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135640946\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135640945-solution\">55<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=10x^4-21x^2+11[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165134058383\" data-depth=\"2\">\n<h3 data-type=\"title\">Numeric<\/h3>\n<p id=\"fs-id1165134058388\">For the following exercises, list all possible rational zeros for the functions.<\/p>\n<div id=\"fs-id1165134058391\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134058392\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=x^4+3x^3-4x+4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165135486016\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135486017\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135486016-solution\">57<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=2x^3+3x^2-8x+5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134204451\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134204452\" data-type=\"problem\">\n<p><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=3x^3+5x^2-5x+4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132920307\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135179860\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165132920307-solution\">59<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=6x^4-10x^2+13x+1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135698642\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135698643\" data-type=\"problem\">\n<p><span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=4x^5-10x^4+8x^3+x^2-8[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165135353045\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<p id=\"fs-id1165135353050\">For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.<\/p>\n<div id=\"fs-id1165135353055\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135353056\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135353055-solution\">61<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=6x^3-7x^2+1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165134431750\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134431751\" data-type=\"problem\">\n<p><span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=4x^3-4x^2-13x-5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135394055\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135394056\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135394055-solution\">63<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=8x^3-6x^2-23x+6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135414275\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135414276\" data-type=\"problem\">\n<p><span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=12x^4+55x^3+12x^2-177x+54[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134469920\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134469921\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134469920-solution\">65<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=16x^4-24x^3+x^2-15x+25[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165134155157\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<p id=\"fs-id1165134155162\">For the following exercises, construct a polynomial function of least degree possible using the given information.<\/p>\n<div id=\"fs-id1165134155167\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134155168\" data-type=\"problem\">\n<p><span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>Real roots: [latex]-1, 1, 3, \\ \\text{and } (2, f(2))=(2, 4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135538724\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135538725\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135538724-solution\">67<\/a><span class=\"os-divider\">. <\/span>Real roots: [latex]-1, 1 \\ \\text{(with multiplicity 2 and 1) and } (2, f(2))=(2, 4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133213938\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133213939\" data-type=\"problem\">\n<p><span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span>Real roots: [latex],2 \\frac{1}{2} \\ \\text{(with multiplicity 2) and } (-3, f(-3))=(-3, 5)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135205845\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135205846\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135205845-solution\">69<\/a><span class=\"os-divider\">. <\/span>Real roots: [latex]-\\frac{1}{2}, 0, \\frac{1}{2} \\ \\text{and } (-2, f(-2))=(-2, 6)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134047574\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134047575\" data-type=\"problem\">\n<p><span class=\"os-number\">70<\/span><span class=\"os-divider\">. <\/span>Real roots: [latex]-4, -1, 1, 4 \\ \\text{and } (-2, f(-2))=(-2, 10)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165134109613\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<p id=\"fs-id1165134109618\">For the following exercises, find the dimensions of the box described.<\/p>\n<div id=\"fs-id1165134043608\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134043610\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134043608-solution\">71<\/a><span class=\"os-divider\">. <\/span>The length is twice as long as the width. The height is 2 inches greater than the width. The volume is 192 cubic inches.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134043619\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134043620\" data-type=\"problem\">\n<p><span class=\"os-number\">72<\/span><span class=\"os-divider\">. <\/span>The length, width, and height are consecutive whole numbers. The volume is 120 cubic inches.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134043625\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134043626\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134043625-solution\">73<\/a><span class=\"os-divider\">. <\/span>The length is one inch more than the width, which is one inch more than the height. The volume is 86.625 cubic inches.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134043635\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134043636\" data-type=\"problem\">\n<p><span class=\"os-number\">74<\/span><span class=\"os-divider\">. <\/span>The length is three times the height and the height is one inch less than the width. The volume is 108 cubic inches.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134043642\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134043643\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134043642-solution\">75<\/a><span class=\"os-divider\">. <\/span>The length is 3 inches more than the width. The width is 2 inches more than the height. The volume is 120 cubic inches.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165134043653\">For the following exercises, find the dimensions of the right circular cylinder described.<\/p>\n<div id=\"fs-id1165134043657\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134043658\" data-type=\"problem\">\n<p><span class=\"os-number\">76<\/span><span class=\"os-divider\">. <\/span>The radius is 3 inches more than the height. The volume is [latex]16\\pi[\/latex] cubic meters.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135315485\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135315486\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135315485-solution\">77<\/a><span class=\"os-divider\">. <\/span>The height is one less than one half the radius. The volume is [latex]72\\pi[\/latex] cubic meters.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135315510\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135315511\" data-type=\"problem\">\n<p><span class=\"os-number\">78<\/span><span class=\"os-divider\">. <\/span>The radius and height differ by one meter. The radius is larger and the volume is [latex]48\\pi[\/latex] cubic meters.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135453003\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135453004\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135453003-solution\">79<\/a><span class=\"os-divider\">. <\/span>The radius and height differ by two meters. The height is greater and the volume is [latex]28.125\\pi[\/latex] cubic meters.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135453028\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135453029\" data-type=\"problem\">\n<p><span class=\"os-number\">80<\/span><span class=\"os-divider\">. <\/span>The radius is [latex]\\frac{1}{3}[\/latex] meter greater than the height. The volume is [latex]\\frac{98}{9}\\pi[\/latex] cubic meters.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n","protected":false},"author":158,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-268","chapter","type-chapter","status-publish","hentry"],"part":158,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/268","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/users\/158"}],"version-history":[{"count":16,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/268\/revisions"}],"predecessor-version":[{"id":1564,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/268\/revisions\/1564"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/158"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/268\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=268"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=268"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=268"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=268"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}