{"id":267,"date":"2025-04-09T17:37:41","date_gmt":"2025-04-09T17:37:41","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/5-4-dividing-polynomials-college-algebra-2e-openstax\/"},"modified":"2025-08-20T21:49:20","modified_gmt":"2025-08-20T21:49:20","slug":"5-4-dividing-polynomials","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/5-4-dividing-polynomials\/","title":{"raw":"5.4 Dividing Polynomials","rendered":"5.4 Dividing Polynomials"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_11f2bd05-8eff-4fb8-9e06-02a657fbb031\" 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>Use long division to divide polynomials.<\/li>\r\n \t<li>Use synthetic division to divide polynomials.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n\r\n[caption id=\"attachment_823\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-823 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-1-300x176.jpeg\" alt=\"\" width=\"300\" height=\"176\" \/> Figure 1. Lincoln Memorial, Washington, D.C. (credit: Ron Cogswell, Flickr)[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165135382145\" class=\"has-noteref\">The exterior of the Lincoln Memorial in Washington, D.C., is a large rectangular solid with length 61.5 meters (m), width 40 m, and height 30 m.[footnote]National Park Service. \"Lincoln Memorial Building Statistics.\" http:\/\/www.nps.gov\/linc\/historyculture\/lincoln-memorial-building-statistics.htm. Accessed 4\/3\/2014[\/footnote]<sup id=\"footnote-ref1\" data-type=\"footnote-number\"><\/sup> We can easily find the volume using elementary geometry.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} V &amp;=&amp; l\\cdot w\\cdot h \\\\ &amp;=&amp;61.5\\cdot 40\\cdot 30 \\\\ &amp;=&amp; 73,800 \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165133214948\">So the volume is 73,800 cubic meters [latex] (m^3). [\/latex] Suppose we knew the volume, length, and width. We could divide to find the height.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} h &amp;=&amp; \\frac{v}{l\\cdot w} \\\\ &amp;=&amp; \\frac{73,800}{61.5\\cdot 40} \\\\ &amp;=&amp; 30 \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165137892463\">As we can confirm from the dimensions above, the height is 30 m. We can use similar methods to find any of the missing dimensions. We can also use the same method if any, or all, of the measurements contain variable expressions. For example, suppose the volume of a rectangular solid is given by the polynomial [latex] 3x^4-3x^3-33x^2+54x. [\/latex] The length of the solid is given by [latex] 3x; [\/latex] the width is given by [latex] x-2. [\/latex] To find the height of the solid, we can use polynomial division, which is the focus of this section.<\/p>\r\n\r\n<section id=\"fs-id1165137676949\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using Long Division to Divide Polynomials<\/h2>\r\n<p id=\"fs-id1165135191647\">We are familiar with the <span id=\"term-00001\" class=\"no-emphasis\" data-type=\"term\">long division<\/span> algorithm for ordinary arithmetic. We begin by dividing into the digits of the dividend that have the greatest place value. We divide, multiply, subtract, include the digit in the next place value position, and repeat. For example, let\u2019s divide 178 by 3 using long division.<\/p>\r\n<span id=\"fs-id1165137564295\" data-type=\"media\" data-alt=\"Steps of long division for intergers.\" data-display=\"block\">\r\n<img class=\" wp-image-824 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-long-division-300x111.jpeg\" alt=\"\" width=\"410\" height=\"152\" \/>\r\n<\/span>\r\n<p id=\"fs-id1165134170235\">Another way to look at the solution is as a sum of parts. This should look familiar, since it is the same method used to check division in elementary arithmetic.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} \\text{dividend} &amp;=&amp; (\\text{divisor}\\cdot \\text{quotient})+\\text{remainder} \\\\ 178 &amp;=&amp; (3\\cdot 59)+1 \\\\ &amp;=&amp; 177+1 \\\\ &amp;=&amp; 178 \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165137640958\">We call this the <strong>Division Algorithm <\/strong>and will discuss it more formally after looking at an example.<\/p>\r\n<p id=\"fs-id1165137933942\">Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials. For example, if we were to divide [latex] 2x^3-3x^2+4x+5 [\/latex] by [latex]\u00a0 x+2[\/latex] using the long division algorithm, it would look like this:<\/p>\r\n<span id=\"fs-id1678300\" data-type=\"media\" data-alt=\"\" data-display=\"block\"><img class=\" wp-image-825 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-lda-300x243.jpeg\" alt=\"\" width=\"632\" height=\"512\" \/>\r\n<\/span>\r\n<p id=\"fs-id1165135191694\">We have found<\/p>\r\n<p style=\"text-align: center;\">[latex] \\frac{2x^3-3x^2+4x+5}{x+2}=2x^2-7x+18-\\frac{31}{x+2} [\/latex]<\/p>\r\nor\r\n<p style=\"text-align: center;\">[latex] 2x^3-3x^2+4x+5=(x+2)(2x^2-7x+18)-31 [\/latex]<\/p>\r\n<p id=\"fs-id1165135181270\">We can identify the dividend, the divisor, the quotient, and the remainder.<\/p>\r\n<span id=\"fs-id1165134164979\" data-type=\"media\" data-alt=\"Identifying the dividend, divisor, quotient and remainder of the polynomial 2x^3-3x^2+4x+5, which is the dividend.\" data-display=\"block\"><img class=\"size-medium wp-image-826 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ddqr-300x61.jpeg\" alt=\"\" width=\"300\" height=\"61\" \/><\/span>\r\n<p id=\"fs-id1165135508592\">Writing the result in this manner illustrates the Division Algorithm.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">The Division Algorithm<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe <strong>Division Algorithm <\/strong>states that, given a polynomial dividend [latex] f(x) [\/latex] and a non-zero polynomial divisor [latex] d(x) [\/latex] where the degree of [latex] d(x0 [\/latex] is less than or equal to the degree of [latex] f(x), [\/latex] there exist unique polynomials [latex] q(x) [\/latex] and [latex] r(x) [\/latex] such that\r\n<p style=\"text-align: center;\">[latex] f(x)=d(x)q(x)+r(x) [\/latex]<\/p>\r\n[latex] q(x) [\/latex] is the quotient and [latex] r(x) [\/latex] is the remainder. The remainder is either equal to zero or has degree strictly less than [latex] d(x). [\/latex]\r\n\r\nIf [latex] r(x)=0, [\/latex] then [latex] d(x) [\/latex] divides evenly into [latex] f(x). [\/latex] This means that, in this case, both [latex] d(x) [\/latex] and [latex] q(x) [\/latex] are factors of [latex] f(x). [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/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 and a binomial, use long division to divide the polynomial by the binomial.<\/strong>\r\n<ol>\r\n \t<li>Set up the division problem.<\/li>\r\n \t<li>Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.<\/li>\r\n \t<li>Multiply the answer by the divisor and write it below the like terms of the dividend.<\/li>\r\n \t<li>Subtract the bottom <span id=\"term-00007\" class=\"no-emphasis\" data-type=\"term\">binomial<\/span> from the top binomial.<\/li>\r\n \t<li>Bring down the next term of the dividend.<\/li>\r\n \t<li>Repeat steps 2\u20135 until reaching the last term of the dividend.<\/li>\r\n \t<li>If the remainder is non-zero, express as a fraction using the divisor as the denominator.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 1: Using Long Division to Divide a Second-Degree Polynomial<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDivide [latex] 5x^2+3x-2 [\/latex] by [latex] x+1. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary><img class=\"wp-image-827 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-1-300x156.jpeg\" alt=\"\" width=\"548\" height=\"285\" \/>\r\n\r\nThe quotient is [latex] 5x-2. [\/latex] The remainder is 0. We write the result as\r\n<p style=\"text-align: center;\">[latex] \\frac{5x^2+3x-2}{x+1}=5x-2 [\/latex]<\/p>\r\nor\r\n<p style=\"text-align: center;\">[latex] 5x^2+3x-2=(x+1)(5x-2) [\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nThis division problem had a remainder of 0. This tells us that the dividend is divided evenly by the divisor, and that the divisor is a factor of the dividend.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 2: Using Long Division to Divide a Third-Degree Polynomial<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDivide [latex] 6x^3+11x^2-31x+15 [\/latex] by [latex] 3x-2. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary><img class=\"wp-image-828 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-2-300x64.jpeg\" alt=\"\" width=\"567\" height=\"121\" \/>\r\n\r\nThere is a remainder of 1. We can express the result as:\r\n<p style=\"text-align: center;\">[latex] \\frac{6x^3+11x^2-31x+15}{3x-2}=2x^2+5x-7+\\frac{1}{3x-2} [\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nWe can check our work by using the Division Algorithm to rewrite the solution. Then multiply.\r\n<p style=\"text-align: center;\">[latex] (3x-2)(2x^2+5x-7)+1=6x^3+11x^2-31x+15 [\/latex]<\/p>\r\nNotice, as we write our result,\r\n<ul>\r\n \t<li>the dividend is [latex] 6x^3+11x^2-31x+15 [\/latex]<\/li>\r\n \t<li>the divisor is [latex] 3x-2 [\/latex]<\/li>\r\n \t<li>the quotient is [latex] 2x^2+5x-7 [\/latex]<\/li>\r\n \t<li>the remainder is 1<\/li>\r\n<\/ul>\r\n<\/details><\/div>\r\n<\/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\nDivide [latex] 16x^3-12x^2+20x-3 [\/latex] by [latex] 4x+5. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165137932621\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using Synthetic Division to Divide Polynomials<\/h2>\r\n<p id=\"fs-id1165137932627\">As we\u2019ve seen, long division of polynomials can involve many steps and be quite cumbersome. <strong>Synthetic division<\/strong> is a shorthand method of dividing polynomials for the special case of dividing by a linear factor whose leading coefficient is 1.<\/p>\r\n<p id=\"fs-id1165137932636\">To illustrate the process, recall the example at the beginning of the section.<\/p>\r\n<p id=\"fs-id1165137932639\">Divide [latex] 2x^3-3x^2+4x+5 [\/latex] by [latex] x+2 [\/latex] using the long division algorithm.<\/p>\r\n<p id=\"fs-id1165135170412\">The final form of the process looked like this:<\/p>\r\n<span id=\"fs-id2502523\" data-type=\"media\" data-alt=\"\" data-display=\"block\">\r\n<img class=\"size-medium wp-image-829 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-synthetic-division-300x185.jpeg\" alt=\"\" width=\"300\" height=\"185\" \/>\r\n<\/span>\r\n<p id=\"fs-id1165137932377\">There is a lot of repetition in the table. If we don\u2019t write the variables but, instead, line up their coefficients in columns under the division sign and also eliminate the partial products, we already have a simpler version of the entire problem.<\/p>\r\n<span id=\"fs-id1165134305375\" data-type=\"media\" data-alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\" data-display=\"block\">\r\n<img class=\"wp-image-830 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-sd-300x68.jpeg\" alt=\"\" width=\"516\" height=\"117\" \/>\r\n<\/span>\r\n<p id=\"fs-id1165134305388\">Synthetic division carries this simplification even a few more steps. Collapse the table by moving each of the rows up to fill any vacant spots. Also, instead of dividing by 2, as we would in division of whole numbers, then multiplying and subtracting the middle product, we change the sign of the \u201cdivisor\u201d to \u20132, multiply and add. The process starts by bringing down the leading coefficient.<\/p>\r\n<span id=\"fs-id1165137696374\" data-type=\"media\" data-alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\" data-display=\"block\">\r\n<img class=\"wp-image-831 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-sd1-300x46.jpeg\" alt=\"\" width=\"574\" height=\"88\" \/>\r\n<\/span>\r\n<p id=\"fs-id1165137696388\">We then multiply it by the \u201cdivisor\u201d and add, repeating this process column by column, until there are no entries left. The bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder. In this case, the quotient is [latex] 2x^2-7x+18 [\/latex] and the remainder is [latex] -31. [\/latex] The process will be made more clear in Example 3.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Synthetic Division<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSynthetic division is a shortcut that can be used when the divisor is a binomial in the form [latex] x-k [\/latex] where [latex] k [\/latex] is a real number. In <strong><span id=\"term-00008\" data-type=\"term\">synthetic division<\/span><\/strong>, only the coefficients are used in the division process.\r\n\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 two polynomials, use synthetic division to divide.<\/strong>\r\n<ol>\r\n \t<li>Write [latex] k [\/latex] for the divisor.<\/li>\r\n \t<li>Write the coefficients of the dividend.<\/li>\r\n \t<li>Bring the lead coefficient down.<\/li>\r\n \t<li>Multiply the lead coefficient by [latex] k. [\/latex] Write the product in the next column.<\/li>\r\n \t<li>Add the terms of the second column.<\/li>\r\n \t<li>Multiply the result by [latex] k. [\/latex] Write the product in the next column.<\/li>\r\n \t<li>Repeat steps 5 and 6 for the remaining columns.<\/li>\r\n \t<li>Use the bottom numbers to write the quotient. The number in the last column is the remainder. The next number from the right has degree 0, the next number has degree 1, and so on.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<section>\r\n<div class=\"os-note-body\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3: Using Synthetic Division to Divide a Second-Degree Polynomial<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse synthetic division to divide [latex] 5x^2-3x-36 [\/latex] by [latex] x-3. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Begin by setting up the synthetic division. Write [latex] k [\/latex] and the coefficients.\r\n\r\n<img class=\"wp-image-832 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3-300x34.jpeg\" alt=\"\" width=\"512\" height=\"58\" \/>\r\n\r\nBring down the lead coefficient. Multiply the lead coefficient by [latex] k. [\/latex]\r\n\r\n<img class=\"wp-image-833 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3.1-300x46.jpeg\" alt=\"\" width=\"443\" height=\"68\" \/>\r\n\r\nContinue by adding the numbers in the second column. Multiply the resulting number by [latex] k. [\/latex] Write the result in the next column. Then add the numbers in the third column.\r\n\r\n<img class=\"wp-image-834 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3.2-300x46.jpeg\" alt=\"\" width=\"528\" height=\"81\" \/>\r\n\r\nThe result is [latex] 5x+12. [\/latex] The remainder is 0. So [latex] x-3 [\/latex] is a factor of the original polynomial.\r\n<h3>Analysis<\/h3>\r\nJust as with long division, we can check our work by multiplying the quotient by the divisor and adding the remainder.\r\n<p style=\"text-align: center;\">[latex] (x-3)(5x+12)+0=5x^2-3x-36 [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 4: Using Synthetic Division to Divide a Third-Degree Polynomial<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse synthetic division to divide [latex] 4x^3+10x^2-6x-20 [\/latex] by [latex] x+2. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The binomial divisor is [latex] x+2 [\/latex] so [latex] k=-2. [\/latex] Add each column, multiply the result by \u20132, and repeat until the last column is reached.\r\n\r\n<img class=\"wp-image-836 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-4-300x46.jpeg\" alt=\"\" width=\"509\" height=\"78\" \/>\r\n\r\nThe result is [latex] 4x^2+2x-10. [\/latex] The remainder is 0. Thus, [latex] x+2 [\/latex] is a factor of [latex] 4x^3+10x^2-6x-20. [\/latex]\r\n<h3>Analysis<\/h3>\r\nThe graph of the polynomial function [latex] f(x)=4x^3+10x^2-6x-20 [\/latex] in Figure 2 shows a zero at [latex] x=k=-2. [\/latex] This confirms that [latex] x+2 [\/latex] is a factor of [latex] 4x^2+10x^2-6x-20. [\/latex]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_837\" align=\"aligncenter\" width=\"241\"]<img class=\"size-medium wp-image-837\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-2-241x300.jpeg\" alt=\"\" width=\"241\" height=\"300\" \/> Figure 2[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 5: Using Synthetic Division to Divide a Fourth-Degree Polynomial<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse synthetic division to divide [latex] -9x^4+10x^3+7x^2-6 [\/latex] by [latex] x-1. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Notice there is no <em data-effect=\"italics\">x<\/em>-term. We will use a zero as the coefficient for that term.\r\n\r\n<img class=\" wp-image-838 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-5.jpeg\" alt=\"\" width=\"237\" height=\"88\" \/>\r\n\r\nThe result is [latex] -9x^3+x^2+8x+8+\\frac{2}{x-1}. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/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 synthetic division to divide [latex] 3x^4+18x^3-3x+40 [\/latex] by [latex] x+7. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135403412\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using Polynomial Division to Solve Application Problems<\/h2>\r\n<p id=\"fs-id1165135403417\">Polynomial division can be used to solve a variety of application problems involving expressions for area and volume. We looked at an application at the beginning of this section. Now we will solve that problem in the following example.<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 6: Using Polynomial Division in an Application Problem<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe volume of a rectangular solid is given by the polynomial [latex] 3x^4-3x^3-33x^2+54x. [\/latex] The length of the solid is given by [latex] 3x [\/latex] and the width is given by [latex] x-2. [\/latex] Find the height, [latex] h, [\/latex] of the solid.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>There are a few ways to approach this problem. We need to divide the expression for the volume of the solid by the expressions for the length and width. Let us create a sketch as in Figure 3.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_839\" align=\"aligncenter\" width=\"338\"]<img class=\" wp-image-839\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-3-300x86.jpeg\" alt=\"\" width=\"338\" height=\"97\" \/> Figure 3[\/caption]\r\n\r\nWe can now write an equation by substituting the known values into the formula for the volume of a rectangular solid.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} V &amp;=&amp; l\\cdot w\\cdot h \\\\ 3x^4-3x^3-33x^2+54x &amp;=&amp; 3x\\cdot (x-2)\\cdot h \\end{array} [\/latex]<\/p>\r\nTo solve for [latex] h, [\/latex] first divide both sides by [latex] 3x. [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} \\frac{3x\\cdot (x-2)\\cdot h}{3x} &amp;=&amp; \\frac{3x^4-3x^3-33x^2+54x}{3x} \\\\ (x-2)h &amp;=&amp; x^3-x^2-11x=18 \\end{array} [\/latex]<\/p>\r\nNow solve for [latex] h [\/latex] using synthetic division.\r\n<p style=\"text-align: center;\">[latex] h=\\frac{x^3-x^2-11x+18}{x-2} [\/latex]<\/p>\r\n<img class=\"size-medium wp-image-840 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-6-300x71.jpeg\" alt=\"\" width=\"300\" height=\"71\" \/>\r\n\r\nThe quotient is [latex] x^2+x-9 [\/latex] and the remainder is 0. The height of the solid is [latex] x^2+x-9. [\/latex]\r\n\r\n<\/details><\/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\nThe area of a rectangle is given by [latex] 3x^3+14x^2-23x+6. [\/latex] The width of the rectangle is given by [latex] x+6. [\/latex] Find an expression for the length of the rectangle.\r\n\r\n<\/div>\r\n<\/div>\r\n<section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_03_05_03\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135694546\" 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 polynomial division.\r\n<ul>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=KUPFg__Djzw\">Dividing a Trinomial by a Binomial Using Long Division<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=chyi4APQJi0\">Dividing a Polynomial by a Binomial Using Long Division<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=_Y0fRbh1RY8\">Ex. 2: Dividing a Polynomial by a Binomial Using Synthetic Division<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=bqm4DoznJxo\">Ex. 4: Dividing a Polynomial by a Binomial Using Synthetic Division<\/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><\/section>\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\">5.4 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1165135255120\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1165135255124\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1165135255129\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135255131\" 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-id1165135255129-solution\">1<\/a><span class=\"os-divider\">. <\/span>If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135443975\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135443976\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>If a polynomial of degree [latex] n [\/latex] is divided by a binomial of degree 1, what is the degree of the quotient?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>\r\n<section id=\"fs-id1165135443995\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id1165135444000\">For the following exercises, use long division to divide. Specify the quotient and the remainder.<\/p>\r\n\r\n<div id=\"fs-id1165135349098\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135349099\" 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-id1165135349098-solution\">3<\/a><span class=\"os-divider\">. <\/span> [latex] (x^2+5x-1)\\div (x-1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165137932684\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137932685\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span> [latex] (2x^2-9x-5)\\div (x-5) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135643143\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135643144\" 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-id1165135643143-solution\">5<\/a><span class=\"os-divider\">. <\/span> [latex] (3x^2+23x+14)\\div (x+7) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135453058\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135453059\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex] (4x^2-10x+6)\\div (4x+2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137833881\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137833882\" 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-id1165137833881-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex] (6x^2-25x-25)\\div (6x+5) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135363204\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135363205\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex] (-x^2-1)\\div (x+1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134472270\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134472271\" 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-id1165134472270-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex] (2x^2-3x+2)\\div (x+2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134352532\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134352533\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex] (x^2-126)\\div (x-5) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135678617\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135678618\" 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-id1165135678617-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex] (3x^2-5x+4)\\div (3x+1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133093355\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133093356\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex] (x^3-3x^2+5x-6)\\div (x-2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135435667\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135435668\" 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-id1165135435667-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex] (2x^3+3x^2-4x+15)\\div (x+3) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135321927\">For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)<\/p>\r\n\r\n<div id=\"fs-id1165135321931\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135321932\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><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 id=\"fs-id1165134474179\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134474180\" 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-id1165134474179-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex] (2x^3-6x^2-7x+6)\\div (x-4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135199519\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137680594\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex] (6x^3-10x^2-7x-15)\\div (x+1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133233060\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133233062\" 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-id1165133233060-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] (4x^3-12x^2-5x-1)\\div (2x+1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133311045\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133311046\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] (9x^3-9x^2+18x+5)\\div (3x-1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135560630\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135560631\" 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-id1165135560630-solution\">19<\/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 id=\"fs-id1165137642766\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137642767\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex] (-6x^3+x^2-4)\\div (2x-3) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135530633\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135530634\" 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-id1165135530633-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex] (2x^3+7x^2-13x-3)\\div (2x-3) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134173742\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134173743\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex] (3x^3-5x^2+2x+3)\\div (x+2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135443762\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135443764\" 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-id1165135443762-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] (4x^3-5x^2+13)\\div (x+4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135403522\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135403523\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex] (x^3-3x+2)\\div (x+2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134061948\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134061949\" 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-id1165134061948-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex] (x^3-21x^2+147x-343)\\div (x-7) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134156069\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134156070\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex] (x^3-15x^2+75x-125)\\div (x-5) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135344106\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135344107\" 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-id1165135344106-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex] (9x^3-x+2)\\div (3x-1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135485710\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135485711\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex] (6x^3-x^2+5x+2)\\div (3x+1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135485136\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135485137\" 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-id1165135485136-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex] (x^4+x^3-3x^2-2x+1)\\div (x+1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134357289\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134357290\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex] (x^4-3x^2+1)\\div (x-1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135390850\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135390852\" 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-id1165135390850-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex] (x^4+2x^3-3x^2+2x+6)\\div (x+3) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137850245\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137850246\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex] (x^4-10x^3+37x^2-60x+36)\\div (x-2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137705401\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137705402\" 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-id1165137705401-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex] (x^4-8x^3+24x^2-32x+16)\\div (x-2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134389913\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134389914\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex] (x^4+5x^3-3x^2-13x+10)\\div (x+5) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134484904\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134484905\" 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-id1165134484904-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex] (x^4-12x^3+54x^2-108x+81)\\div (x-3) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137696095\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137696096\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex] (4x^4-2x^3-4x+2)\\div (2x-1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135575253\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135575254\" 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-id1165135575253-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex] (4x^4+2x^3-4x^2+2x+2)\\div (2x+1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135353027\">For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.<\/p>\r\n\r\n<div id=\"fs-id1165135353032\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135353033\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex] x-2, 4x^3-3x^2-8x+4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134031382\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134031383\" 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-id1165134031382-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex] x-2, 3x^4-6x^3-5x+10 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135240972\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135240974\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex] x+3, -4x^3+5x^2+8 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135394052\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135394053\" 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-id1165135394052-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex] x-2, 4x^4-15x^2-4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137851216\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135397213\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex] x-\\frac{1}{2}, 2x^4-x^3+2x-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135547391\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135547392\" 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-id1165135547391-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex] x+\\frac{1}{3}, 3x^4+x^3-3x+1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165135481299\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphical<\/h3>\r\n<p id=\"fs-id1165135481304\">For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.<\/p>\r\n\r\n<div id=\"fs-id1165135436617\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135436618\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> Factor is [latex] x^2-x+3 [\/latex]\r\n\r\n<img class=\"alignnone size-medium wp-image-841\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.44-300x242.jpeg\" alt=\"\" width=\"300\" height=\"242\" \/>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165135423510\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135423511\" 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-id1165135423510-solution\">45<\/a><span class=\"os-divider\">. <\/span> Factor is [latex] x^2+2x+4 [\/latex]\r\n\r\n<img class=\"alignnone size-medium wp-image-842\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.45-300x242.jpeg\" alt=\"\" width=\"300\" height=\"242\" \/>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135412915\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135412916\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> Factor is [latex] x^2+2x+5 [\/latex]\r\n\r\n<img class=\"alignnone size-medium wp-image-843\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.46-300x242.jpeg\" alt=\"\" width=\"300\" height=\"242\" \/>\r\n\r\n<span id=\"fs-id1165135176349\" data-type=\"media\" data-alt=\"Graph of a polynomial that has a x-intercept at 2.\"><\/span><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135176362-solution\">47<\/a><span class=\"os-divider\">. <\/span>Factor is [latex] x^2+x+1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135176362\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135176363\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n\r\n<img class=\"alignnone size-medium wp-image-844\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.47-300x261.jpeg\" alt=\"\" width=\"300\" height=\"261\" \/>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134089490\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134089491\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> Factor is [latex] x^2+2x+2 [\/latex]\r\n\r\n<img class=\"alignnone size-medium wp-image-845\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.48-300x193.jpeg\" alt=\"\" width=\"300\" height=\"193\" \/>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137888975\">For the following exercises, use synthetic division to find the quotient and remainder.<\/p>\r\n\r\n<div id=\"fs-id1165137888978\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137888979\" 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-id1165137888978-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex] \\frac{4x^3-33}{x-2} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135209652\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135209653\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex]- \\frac{2x^3+25}{x+3} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135443854\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135443855\" 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-id1165135443854-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex] \\frac{3x^3+2x-5}{x-1} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135154363\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135154364\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex] \\frac{-4x^2-x^2-12}{x+4} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135155437\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135155438\" 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-id1165135155437-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex] \\frac{x^4-22}{x+2} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165135501101\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<p id=\"fs-id1165135501106\">For the following exercises, use a calculator with CAS to answer the questions.<\/p>\r\n\r\n<div id=\"fs-id1165137889809\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137889810\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>Consider [latex] \\frac{x^k-1}{x-1} [\/latex] with [latex] k=1, 2, 3. [\/latex] What do you expect the result to be if [latex] k=4? [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135534969\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135534970\" 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-id1165135534969-solution\">55<\/a><span class=\"os-divider\">. <\/span>Consider [latex] \\frac{x^k+1}{x+1} [\/latex] for [latex] k=1, 3, 5. [\/latex] What do you expect the result to be if [latex] k=7? [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135241323\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135241324\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>Consider [latex] \\frac{x^4-k^4}{x-k} [\/latex] for [latex] k=1, 2, 3. [\/latex] What do you expect the result to be if [latex] k=4? [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132939184\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132939185\" 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-id1165132939184-solution\">57<\/a><span class=\"os-divider\">. <\/span>Consider [latex] \\frac{x^k}{x+1} [\/latex] with [latex] k=1, 2, 3. [\/latex] What do you expect the result to be if [latex] k=4? [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134261659\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134261660\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>Consider [latex] \\frac{x^k}{x-1} [\/latex] with [latex] k=1, 2, 3. [\/latex] What do you expect the result to be if [latex] k=4? [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165135191971\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<p id=\"fs-id1165135191976\">For the following exercises, use synthetic division to determine the quotient involving a complex number.<\/p>\r\n\r\n<div id=\"fs-id1165135191981\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135191982\" 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-id1165135191981-solution\">59<\/a><span class=\"os-divider\">. <\/span> [latex] \\frac{x+1}{x-i} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165135593133\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135593134\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span> [latex] \\frac{x^2+1}{x-i} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135593178\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135593179\" 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-id1165135593178-solution\">61<\/a><span class=\"os-divider\">. <\/span> [latex] \\frac{x+1}{x+i} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137777664\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137777665\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span> [latex] \\frac{x^2+1}{x+i} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134248021\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134248022\" 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-id1165134248021-solution\">63<\/a><span class=\"os-divider\">. <\/span> [latex] \\frac{x^3+1}{x-i} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165135180069\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<p id=\"fs-id1165135180074\">For the following exercises, use the given length and area of a rectangle to express the width algebraically.<\/p>\r\n\r\n<div id=\"fs-id1165135180078\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135440138\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>Length is [latex] x+5, [\/latex] area is [latex] 2x^2+9x-5. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135440195\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135440196\" 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-id1165135440195-solution\">65<\/a><span class=\"os-divider\">. <\/span>Length is [latex] 2x+5, [\/latex] area is [latex] 4x^3+10x^2+6x+15. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135264719\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135264720\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>Length is [latex] 3x-4, [\/latex] area is [latex] 6x^4-8x^3+9x^2-9x-4. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135339554\">For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.<\/p>\r\n\r\n<div id=\"fs-id1165135339559\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135339560\" 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-id1165135339559-solution\">67<\/a><span class=\"os-divider\">. <\/span>Volume is [latex] 12x^3+20x^2-21x-36, [\/latex] length is [latex] 2x+3, [\/latex] width is [latex] 3x-4. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134071557\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134071558\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span>Volume is [latex] 18x^3-21x^2-40x+48, [\/latex] length is [latex] 3x-4, [\/latex] width is [latex] 3x-4. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132920308\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132920309\" 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-id1165132920308-solution\">69<\/a><span class=\"os-divider\">. <\/span>Volume is [latex] 10x^3+27x^2+2x-24, [\/latex] length is [latex] 5x-4, [\/latex] width is [latex] 2x+3. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135564178\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135564179\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">70<\/span><span class=\"os-divider\">. <\/span>Volume is [latex] 10x^3+30x^2-8x-24, [\/latex] length is [latex] 2, [\/latex] width is [latex] x+3. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135514530\">For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.<\/p>\r\n\r\n<div id=\"fs-id1165135514534\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135514535\" 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-id1165135514534-solution\">71<\/a><span class=\"os-divider\">. <\/span>Volume is [latex] \\pi 925x^3-65x^2-29x-3), [\/latex] radius is [latex] 5x+1. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134031328\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134031329\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">72. <\/span>Volume is [latex] \\pi (4x^3+12x^2-15x-50), [\/latex] radius is [latex] 2x+5. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134258601\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134258602\" 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-id1165134258601-solution\">73<\/a><span class=\"os-divider\">. <\/span>Volume is [latex] \\pi (3x^4+24x^3+46x^2-16x-32), [\/latex] radius is [latex] x+4. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_11f2bd05-8eff-4fb8-9e06-02a657fbb031\" 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>Use long division to divide polynomials.<\/li>\n<li>Use synthetic division to divide polynomials.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<figure id=\"attachment_823\" aria-describedby=\"caption-attachment-823\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-823 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-1-300x176.jpeg\" alt=\"\" width=\"300\" height=\"176\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-1-300x176.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-1-65x38.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-1-225x132.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-1-350x205.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-1.jpeg 488w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-823\" class=\"wp-caption-text\">Figure 1. Lincoln Memorial, Washington, D.C. (credit: Ron Cogswell, Flickr)<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165135382145\" class=\"has-noteref\">The exterior of the Lincoln Memorial in Washington, D.C., is a large rectangular solid with length 61.5 meters (m), width 40 m, and height 30 m.<a class=\"footnote\" title=\"National Park Service. &quot;Lincoln Memorial Building Statistics.&quot; http:\/\/www.nps.gov\/linc\/historyculture\/lincoln-memorial-building-statistics.htm. Accessed 4\/3\/2014\" id=\"return-footnote-267-1\" href=\"#footnote-267-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><sup id=\"footnote-ref1\" data-type=\"footnote-number\"><\/sup> We can easily find the volume using elementary geometry.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} V &=& l\\cdot w\\cdot h \\\\ &=&61.5\\cdot 40\\cdot 30 \\\\ &=& 73,800 \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165133214948\">So the volume is 73,800 cubic meters [latex](m^3).[\/latex] Suppose we knew the volume, length, and width. We could divide to find the height.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} h &=& \\frac{v}{l\\cdot w} \\\\ &=& \\frac{73,800}{61.5\\cdot 40} \\\\ &=& 30 \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165137892463\">As we can confirm from the dimensions above, the height is 30 m. We can use similar methods to find any of the missing dimensions. We can also use the same method if any, or all, of the measurements contain variable expressions. For example, suppose the volume of a rectangular solid is given by the polynomial [latex]3x^4-3x^3-33x^2+54x.[\/latex] The length of the solid is given by [latex]3x;[\/latex] the width is given by [latex]x-2.[\/latex] To find the height of the solid, we can use polynomial division, which is the focus of this section.<\/p>\n<section id=\"fs-id1165137676949\" data-depth=\"1\">\n<h2 data-type=\"title\">Using Long Division to Divide Polynomials<\/h2>\n<p id=\"fs-id1165135191647\">We are familiar with the <span id=\"term-00001\" class=\"no-emphasis\" data-type=\"term\">long division<\/span> algorithm for ordinary arithmetic. We begin by dividing into the digits of the dividend that have the greatest place value. We divide, multiply, subtract, include the digit in the next place value position, and repeat. For example, let\u2019s divide 178 by 3 using long division.<\/p>\n<p><span id=\"fs-id1165137564295\" data-type=\"media\" data-alt=\"Steps of long division for intergers.\" data-display=\"block\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-824 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-long-division-300x111.jpeg\" alt=\"\" width=\"410\" height=\"152\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-long-division-300x111.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-long-division-65x24.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-long-division-225x84.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-long-division-350x130.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-long-division.jpeg 487w\" sizes=\"auto, (max-width: 410px) 100vw, 410px\" \/><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165134170235\">Another way to look at the solution is as a sum of parts. This should look familiar, since it is the same method used to check division in elementary arithmetic.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} \\text{dividend} &=& (\\text{divisor}\\cdot \\text{quotient})+\\text{remainder} \\\\ 178 &=& (3\\cdot 59)+1 \\\\ &=& 177+1 \\\\ &=& 178 \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165137640958\">We call this the <strong>Division Algorithm <\/strong>and will discuss it more formally after looking at an example.<\/p>\n<p id=\"fs-id1165137933942\">Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials. For example, if we were to divide [latex]2x^3-3x^2+4x+5[\/latex] by [latex]\u00a0 x+2[\/latex] using the long division algorithm, it would look like this:<\/p>\n<p><span id=\"fs-id1678300\" data-type=\"media\" data-alt=\"\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-825 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-lda-300x243.jpeg\" alt=\"\" width=\"632\" height=\"512\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-lda-300x243.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-lda-65x53.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-lda-225x183.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-lda-350x284.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-lda.jpeg 731w\" sizes=\"auto, (max-width: 632px) 100vw, 632px\" \/><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165135191694\">We have found<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{2x^3-3x^2+4x+5}{x+2}=2x^2-7x+18-\\frac{31}{x+2}[\/latex]<\/p>\n<p>or<\/p>\n<p style=\"text-align: center;\">[latex]2x^3-3x^2+4x+5=(x+2)(2x^2-7x+18)-31[\/latex]<\/p>\n<p id=\"fs-id1165135181270\">We can identify the dividend, the divisor, the quotient, and the remainder.<\/p>\n<p><span id=\"fs-id1165134164979\" data-type=\"media\" data-alt=\"Identifying the dividend, divisor, quotient and remainder of the polynomial 2x^3-3x^2+4x+5, which is the dividend.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-826 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ddqr-300x61.jpeg\" alt=\"\" width=\"300\" height=\"61\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ddqr-300x61.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ddqr-65x13.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ddqr-225x46.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ddqr-350x71.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ddqr.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/span><\/p>\n<p id=\"fs-id1165135508592\">Writing the result in this manner illustrates the Division Algorithm.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">The Division Algorithm<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The <strong>Division Algorithm <\/strong>states that, given a polynomial dividend [latex]f(x)[\/latex] and a non-zero polynomial divisor [latex]d(x)[\/latex] where the degree of [latex]d(x0[\/latex] is less than or equal to the degree of [latex]f(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>[latex]q(x)[\/latex] is the quotient and [latex]r(x)[\/latex] is the remainder. The remainder is either equal to zero or has degree strictly less than [latex]d(x).[\/latex]<\/p>\n<p>If [latex]r(x)=0,[\/latex] then [latex]d(x)[\/latex] divides evenly into [latex]f(x).[\/latex] This means that, in this case, both [latex]d(x)[\/latex] and [latex]q(x)[\/latex] are factors of [latex]f(x).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\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 and a binomial, use long division to divide the polynomial by the binomial.<\/strong><\/p>\n<ol>\n<li>Set up the division problem.<\/li>\n<li>Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.<\/li>\n<li>Multiply the answer by the divisor and write it below the like terms of the dividend.<\/li>\n<li>Subtract the bottom <span id=\"term-00007\" class=\"no-emphasis\" data-type=\"term\">binomial<\/span> from the top binomial.<\/li>\n<li>Bring down the next term of the dividend.<\/li>\n<li>Repeat steps 2\u20135 until reaching the last term of the dividend.<\/li>\n<li>If the remainder is non-zero, express as a fraction using the divisor as the denominator.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1: Using Long Division to Divide a Second-Degree Polynomial<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Divide [latex]5x^2+3x-2[\/latex] by [latex]x+1.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-827 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-1-300x156.jpeg\" alt=\"\" width=\"548\" height=\"285\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-1-300x156.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-1-65x34.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-1-225x117.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-1-350x182.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-1.jpeg 731w\" sizes=\"auto, (max-width: 548px) 100vw, 548px\" \/><\/p>\n<p>The quotient is [latex]5x-2.[\/latex] The remainder is 0. We write the result as<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{5x^2+3x-2}{x+1}=5x-2[\/latex]<\/p>\n<p>or<\/p>\n<p style=\"text-align: center;\">[latex]5x^2+3x-2=(x+1)(5x-2)[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>This division problem had a remainder of 0. This tells us that the dividend is divided evenly by the divisor, and that the divisor is a factor of the dividend.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Using Long Division to Divide a Third-Degree Polynomial<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Divide [latex]6x^3+11x^2-31x+15[\/latex] by [latex]3x-2.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-828 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-2-300x64.jpeg\" alt=\"\" width=\"567\" height=\"121\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-2-300x64.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-2-768x164.jpeg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-2-65x14.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-2-225x48.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-2-350x75.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-2.jpeg 975w\" sizes=\"auto, (max-width: 567px) 100vw, 567px\" \/><\/p>\n<p>There is a remainder of 1. We can express the result as:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{6x^3+11x^2-31x+15}{3x-2}=2x^2+5x-7+\\frac{1}{3x-2}[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>We can check our work by using the Division Algorithm to rewrite the solution. Then multiply.<\/p>\n<p style=\"text-align: center;\">[latex](3x-2)(2x^2+5x-7)+1=6x^3+11x^2-31x+15[\/latex]<\/p>\n<p>Notice, as we write our result,<\/p>\n<ul>\n<li>the dividend is [latex]6x^3+11x^2-31x+15[\/latex]<\/li>\n<li>the divisor is [latex]3x-2[\/latex]<\/li>\n<li>the quotient is [latex]2x^2+5x-7[\/latex]<\/li>\n<li>the remainder is 1<\/li>\n<\/ul>\n<\/details>\n<\/div>\n<\/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>Divide [latex]16x^3-12x^2+20x-3[\/latex] by [latex]4x+5.[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165137932621\" data-depth=\"1\">\n<h2 data-type=\"title\">Using Synthetic Division to Divide Polynomials<\/h2>\n<p id=\"fs-id1165137932627\">As we\u2019ve seen, long division of polynomials can involve many steps and be quite cumbersome. <strong>Synthetic division<\/strong> is a shorthand method of dividing polynomials for the special case of dividing by a linear factor whose leading coefficient is 1.<\/p>\n<p id=\"fs-id1165137932636\">To illustrate the process, recall the example at the beginning of the section.<\/p>\n<p id=\"fs-id1165137932639\">Divide [latex]2x^3-3x^2+4x+5[\/latex] by [latex]x+2[\/latex] using the long division algorithm.<\/p>\n<p id=\"fs-id1165135170412\">The final form of the process looked like this:<\/p>\n<p><span id=\"fs-id2502523\" data-type=\"media\" data-alt=\"\" data-display=\"block\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-829 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-synthetic-division-300x185.jpeg\" alt=\"\" width=\"300\" height=\"185\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-synthetic-division-300x185.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-synthetic-division-65x40.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-synthetic-division-225x138.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-synthetic-division.jpeg 325w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165137932377\">There is a lot of repetition in the table. If we don\u2019t write the variables but, instead, line up their coefficients in columns under the division sign and also eliminate the partial products, we already have a simpler version of the entire problem.<\/p>\n<p><span id=\"fs-id1165134305375\" data-type=\"media\" data-alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\" data-display=\"block\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-830 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-sd-300x68.jpeg\" alt=\"\" width=\"516\" height=\"117\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-sd-300x68.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-sd-65x15.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-sd-225x51.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-sd-350x79.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-sd.jpeg 487w\" sizes=\"auto, (max-width: 516px) 100vw, 516px\" \/><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165134305388\">Synthetic division carries this simplification even a few more steps. Collapse the table by moving each of the rows up to fill any vacant spots. Also, instead of dividing by 2, as we would in division of whole numbers, then multiplying and subtracting the middle product, we change the sign of the \u201cdivisor\u201d to \u20132, multiply and add. The process starts by bringing down the leading coefficient.<\/p>\n<p><span id=\"fs-id1165137696374\" data-type=\"media\" data-alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\" data-display=\"block\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-831 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-sd1-300x46.jpeg\" alt=\"\" width=\"574\" height=\"88\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-sd1-300x46.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-sd1-65x10.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-sd1-225x34.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-sd1-350x53.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-sd1.jpeg 487w\" sizes=\"auto, (max-width: 574px) 100vw, 574px\" \/><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165137696388\">We then multiply it by the \u201cdivisor\u201d and add, repeating this process column by column, until there are no entries left. The bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder. In this case, the quotient is [latex]2x^2-7x+18[\/latex] and the remainder is [latex]-31.[\/latex] The process will be made more clear in Example 3.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Synthetic Division<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Synthetic division is a shortcut that can be used when the divisor is a binomial in the form [latex]x-k[\/latex] where [latex]k[\/latex] is a real number. In <strong><span id=\"term-00008\" data-type=\"term\">synthetic division<\/span><\/strong>, only the coefficients are used in the division process.<\/p>\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 two polynomials, use synthetic division to divide.<\/strong><\/p>\n<ol>\n<li>Write [latex]k[\/latex] for the divisor.<\/li>\n<li>Write the coefficients of the dividend.<\/li>\n<li>Bring the lead coefficient down.<\/li>\n<li>Multiply the lead coefficient by [latex]k.[\/latex] Write the product in the next column.<\/li>\n<li>Add the terms of the second column.<\/li>\n<li>Multiply the result by [latex]k.[\/latex] Write the product in the next column.<\/li>\n<li>Repeat steps 5 and 6 for the remaining columns.<\/li>\n<li>Use the bottom numbers to write the quotient. The number in the last column is the remainder. The next number from the right has degree 0, the next number has degree 1, and so on.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<section>\n<div class=\"os-note-body\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Using Synthetic Division to Divide a Second-Degree Polynomial<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use synthetic division to divide [latex]5x^2-3x-36[\/latex] by [latex]x-3.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Begin by setting up the synthetic division. Write [latex]k[\/latex] and the coefficients.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-832 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3-300x34.jpeg\" alt=\"\" width=\"512\" height=\"58\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3-300x34.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3-65x7.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3-225x25.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3-350x40.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3.jpeg 487w\" sizes=\"auto, (max-width: 512px) 100vw, 512px\" \/><\/p>\n<p>Bring down the lead coefficient. Multiply the lead coefficient by [latex]k.[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-833 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3.1-300x46.jpeg\" alt=\"\" width=\"443\" height=\"68\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3.1-300x46.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3.1-65x10.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3.1-225x34.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3.1-350x53.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3.1.jpeg 487w\" sizes=\"auto, (max-width: 443px) 100vw, 443px\" \/><\/p>\n<p>Continue by adding the numbers in the second column. Multiply the resulting number by [latex]k.[\/latex] Write the result in the next column. Then add the numbers in the third column.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-834 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3.2-300x46.jpeg\" alt=\"\" width=\"528\" height=\"81\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3.2-300x46.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3.2-65x10.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3.2-225x34.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3.2-350x53.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-3.2.jpeg 487w\" sizes=\"auto, (max-width: 528px) 100vw, 528px\" \/><\/p>\n<p>The result is [latex]5x+12.[\/latex] The remainder is 0. So [latex]x-3[\/latex] is a factor of the original polynomial.<\/p>\n<h3>Analysis<\/h3>\n<p>Just as with long division, we can check our work by multiplying the quotient by the divisor and adding the remainder.<\/p>\n<p style=\"text-align: center;\">[latex](x-3)(5x+12)+0=5x^2-3x-36[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Using Synthetic Division to Divide a Third-Degree Polynomial<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use synthetic division to divide [latex]4x^3+10x^2-6x-20[\/latex] by [latex]x+2.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The binomial divisor is [latex]x+2[\/latex] so [latex]k=-2.[\/latex] Add each column, multiply the result by \u20132, and repeat until the last column is reached.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-836 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-4-300x46.jpeg\" alt=\"\" width=\"509\" height=\"78\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-4-300x46.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-4-65x10.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-4-225x34.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-4-350x53.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-4.jpeg 487w\" sizes=\"auto, (max-width: 509px) 100vw, 509px\" \/><\/p>\n<p>The result is [latex]4x^2+2x-10.[\/latex] The remainder is 0. Thus, [latex]x+2[\/latex] is a factor of [latex]4x^3+10x^2-6x-20.[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>The graph of the polynomial function [latex]f(x)=4x^3+10x^2-6x-20[\/latex] in Figure 2 shows a zero at [latex]x=k=-2.[\/latex] This confirms that [latex]x+2[\/latex] is a factor of [latex]4x^2+10x^2-6x-20.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_837\" aria-describedby=\"caption-attachment-837\" style=\"width: 241px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-837\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-2-241x300.jpeg\" alt=\"\" width=\"241\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-2-241x300.jpeg 241w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-2-65x81.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-2-225x280.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-2-350x435.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-2.jpeg 425w\" sizes=\"auto, (max-width: 241px) 100vw, 241px\" \/><figcaption id=\"caption-attachment-837\" class=\"wp-caption-text\">Figure 2<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 5: Using Synthetic Division to Divide a Fourth-Degree Polynomial<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use synthetic division to divide [latex]-9x^4+10x^3+7x^2-6[\/latex] by [latex]x-1.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Notice there is no <em data-effect=\"italics\">x<\/em>-term. We will use a zero as the coefficient for that term.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-838 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-5.jpeg\" alt=\"\" width=\"237\" height=\"88\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-5.jpeg 205w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-5-65x24.jpeg 65w\" sizes=\"auto, (max-width: 237px) 100vw, 237px\" \/><\/p>\n<p>The result is [latex]-9x^3+x^2+8x+8+\\frac{2}{x-1}.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/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 synthetic division to divide [latex]3x^4+18x^3-3x+40[\/latex] by [latex]x+7.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135403412\" data-depth=\"1\">\n<h2 data-type=\"title\">Using Polynomial Division to Solve Application Problems<\/h2>\n<p id=\"fs-id1165135403417\">Polynomial division can be used to solve a variety of application problems involving expressions for area and volume. We looked at an application at the beginning of this section. Now we will solve that problem in the following example.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 6: Using Polynomial Division in an Application Problem<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The volume of a rectangular solid is given by the polynomial [latex]3x^4-3x^3-33x^2+54x.[\/latex] The length of the solid is given by [latex]3x[\/latex] and the width is given by [latex]x-2.[\/latex] Find the height, [latex]h,[\/latex] of the solid.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>There are a few ways to approach this problem. We need to divide the expression for the volume of the solid by the expressions for the length and width. Let us create a sketch as in Figure 3.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_839\" aria-describedby=\"caption-attachment-839\" style=\"width: 338px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-839\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-3-300x86.jpeg\" alt=\"\" width=\"338\" height=\"97\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-3-300x86.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-3-65x19.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-3-225x65.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-3-350x101.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-fig-3.jpeg 487w\" sizes=\"auto, (max-width: 338px) 100vw, 338px\" \/><figcaption id=\"caption-attachment-839\" class=\"wp-caption-text\">Figure 3<\/figcaption><\/figure>\n<p>We can now write an equation by substituting the known values into the formula for the volume of a rectangular solid.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} V &=& l\\cdot w\\cdot h \\\\ 3x^4-3x^3-33x^2+54x &=& 3x\\cdot (x-2)\\cdot h \\end{array}[\/latex]<\/p>\n<p>To solve for [latex]h,[\/latex] first divide both sides by [latex]3x.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} \\frac{3x\\cdot (x-2)\\cdot h}{3x} &=& \\frac{3x^4-3x^3-33x^2+54x}{3x} \\\\ (x-2)h &=& x^3-x^2-11x=18 \\end{array}[\/latex]<\/p>\n<p>Now solve for [latex]h[\/latex] using synthetic division.<\/p>\n<p style=\"text-align: center;\">[latex]h=\\frac{x^3-x^2-11x+18}{x-2}[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-840 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-6-300x71.jpeg\" alt=\"\" width=\"300\" height=\"71\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-6-300x71.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-6-65x15.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-6-225x53.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4-ex-6.jpeg 325w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>The quotient is [latex]x^2+x-9[\/latex] and the remainder is 0. The height of the solid is [latex]x^2+x-9.[\/latex]<\/p>\n<\/details>\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>The area of a rectangle is given by [latex]3x^3+14x^2-23x+6.[\/latex] The width of the rectangle is given by [latex]x+6.[\/latex] Find an expression for the length of the rectangle.<\/p>\n<\/div>\n<\/div>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_03_05_03\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135694546\" 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 polynomial division.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=KUPFg__Djzw\">Dividing a Trinomial by a Binomial Using Long Division<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=chyi4APQJi0\">Dividing a Polynomial by a Binomial Using Long Division<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=_Y0fRbh1RY8\">Ex. 2: Dividing a Polynomial by a Binomial Using Synthetic Division<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=bqm4DoznJxo\">Ex. 4: Dividing a Polynomial by a Binomial Using Synthetic Division<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\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\">5.4 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1165135255120\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1165135255124\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1165135255129\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135255131\" 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-id1165135255129-solution\">1<\/a><span class=\"os-divider\">. <\/span>If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135443975\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135443976\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>If a polynomial of degree [latex]n[\/latex] is divided by a binomial of degree 1, what is the degree of the quotient?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<section id=\"fs-id1165135443995\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id1165135444000\">For the following exercises, use long division to divide. Specify the quotient and the remainder.<\/p>\n<div id=\"fs-id1165135349098\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135349099\" 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-id1165135349098-solution\">3<\/a><span class=\"os-divider\">. <\/span> [latex](x^2+5x-1)\\div (x-1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165137932684\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137932685\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span> [latex](2x^2-9x-5)\\div (x-5)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135643143\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135643144\" 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-id1165135643143-solution\">5<\/a><span class=\"os-divider\">. <\/span> [latex](3x^2+23x+14)\\div (x+7)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135453058\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135453059\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex](4x^2-10x+6)\\div (4x+2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137833881\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137833882\" 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-id1165137833881-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex](6x^2-25x-25)\\div (6x+5)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135363204\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135363205\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex](-x^2-1)\\div (x+1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134472270\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134472271\" 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-id1165134472270-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex](2x^2-3x+2)\\div (x+2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134352532\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134352533\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex](x^2-126)\\div (x-5)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135678617\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135678618\" 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-id1165135678617-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex](3x^2-5x+4)\\div (3x+1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133093355\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133093356\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex](x^3-3x^2+5x-6)\\div (x-2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135435667\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135435668\" 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-id1165135435667-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex](2x^3+3x^2-4x+15)\\div (x+3)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135321927\">For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)<\/p>\n<div id=\"fs-id1165135321931\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135321932\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><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 id=\"fs-id1165134474179\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134474180\" 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-id1165134474179-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex](2x^3-6x^2-7x+6)\\div (x-4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135199519\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137680594\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex](6x^3-10x^2-7x-15)\\div (x+1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133233060\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133233062\" 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-id1165133233060-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex](4x^3-12x^2-5x-1)\\div (2x+1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133311045\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133311046\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex](9x^3-9x^2+18x+5)\\div (3x-1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135560630\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135560631\" 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-id1165135560630-solution\">19<\/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 id=\"fs-id1165137642766\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137642767\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex](-6x^3+x^2-4)\\div (2x-3)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135530633\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135530634\" 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-id1165135530633-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex](2x^3+7x^2-13x-3)\\div (2x-3)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134173742\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134173743\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex](3x^3-5x^2+2x+3)\\div (x+2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135443762\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135443764\" 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-id1165135443762-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex](4x^3-5x^2+13)\\div (x+4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135403522\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135403523\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex](x^3-3x+2)\\div (x+2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134061948\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134061949\" 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-id1165134061948-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex](x^3-21x^2+147x-343)\\div (x-7)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134156069\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134156070\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex](x^3-15x^2+75x-125)\\div (x-5)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135344106\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135344107\" 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-id1165135344106-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex](9x^3-x+2)\\div (3x-1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135485710\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135485711\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex](6x^3-x^2+5x+2)\\div (3x+1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135485136\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135485137\" 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-id1165135485136-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex](x^4+x^3-3x^2-2x+1)\\div (x+1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134357289\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134357290\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex](x^4-3x^2+1)\\div (x-1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135390850\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135390852\" 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-id1165135390850-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex](x^4+2x^3-3x^2+2x+6)\\div (x+3)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137850245\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137850246\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex](x^4-10x^3+37x^2-60x+36)\\div (x-2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137705401\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137705402\" 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-id1165137705401-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex](x^4-8x^3+24x^2-32x+16)\\div (x-2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134389913\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134389914\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex](x^4+5x^3-3x^2-13x+10)\\div (x+5)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134484904\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134484905\" 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-id1165134484904-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex](x^4-12x^3+54x^2-108x+81)\\div (x-3)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137696095\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137696096\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex](4x^4-2x^3-4x+2)\\div (2x-1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135575253\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135575254\" 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-id1165135575253-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex](4x^4+2x^3-4x^2+2x+2)\\div (2x+1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135353027\">For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.<\/p>\n<div id=\"fs-id1165135353032\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135353033\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex]x-2, 4x^3-3x^2-8x+4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134031382\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134031383\" 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-id1165134031382-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex]x-2, 3x^4-6x^3-5x+10[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135240972\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135240974\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex]x+3, -4x^3+5x^2+8[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135394052\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135394053\" 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-id1165135394052-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex]x-2, 4x^4-15x^2-4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137851216\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135397213\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex]x-\\frac{1}{2}, 2x^4-x^3+2x-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135547391\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135547392\" 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-id1165135547391-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex]x+\\frac{1}{3}, 3x^4+x^3-3x+1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165135481299\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphical<\/h3>\n<p id=\"fs-id1165135481304\">For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.<\/p>\n<div id=\"fs-id1165135436617\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135436618\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> Factor is [latex]x^2-x+3[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-841\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.44-300x242.jpeg\" alt=\"\" width=\"300\" height=\"242\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.44-300x242.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.44-65x53.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.44-225x182.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.44.jpeg 312w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165135423510\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135423511\" 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-id1165135423510-solution\">45<\/a><span class=\"os-divider\">. <\/span> Factor is [latex]x^2+2x+4[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-842\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.45-300x242.jpeg\" alt=\"\" width=\"300\" height=\"242\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.45-300x242.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.45-65x52.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.45-225x181.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.45.jpeg 313w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135412915\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135412916\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> Factor is [latex]x^2+2x+5[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-843\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.46-300x242.jpeg\" alt=\"\" width=\"300\" height=\"242\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.46-300x242.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.46-65x52.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.46-225x181.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.46.jpeg 313w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><span id=\"fs-id1165135176349\" data-type=\"media\" data-alt=\"Graph of a polynomial that has a x-intercept at 2.\"><\/span><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135176362-solution\">47<\/a><span class=\"os-divider\">. <\/span>Factor is [latex]x^2+x+1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135176362\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135176363\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-844\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.47-300x261.jpeg\" alt=\"\" width=\"300\" height=\"261\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.47-300x261.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.47-65x57.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.47-225x196.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.47-350x305.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.47.jpeg 465w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134089490\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134089491\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> Factor is [latex]x^2+2x+2[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-845\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.48-300x193.jpeg\" alt=\"\" width=\"300\" height=\"193\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.48-300x193.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.48-65x42.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.48-225x145.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.48-350x226.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.4.48.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137888975\">For the following exercises, use synthetic division to find the quotient and remainder.<\/p>\n<div id=\"fs-id1165137888978\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137888979\" 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-id1165137888978-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex]\\frac{4x^3-33}{x-2}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135209652\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135209653\" data-type=\"problem\">\n<p><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex]- \\frac{2x^3+25}{x+3}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135443854\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135443855\" 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-id1165135443854-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex]\\frac{3x^3+2x-5}{x-1}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135154363\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135154364\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex]\\frac{-4x^2-x^2-12}{x+4}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135155437\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135155438\" 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-id1165135155437-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex]\\frac{x^4-22}{x+2}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165135501101\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<p id=\"fs-id1165135501106\">For the following exercises, use a calculator with CAS to answer the questions.<\/p>\n<div id=\"fs-id1165137889809\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137889810\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>Consider [latex]\\frac{x^k-1}{x-1}[\/latex] with [latex]k=1, 2, 3.[\/latex] What do you expect the result to be if [latex]k=4?[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135534969\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135534970\" 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-id1165135534969-solution\">55<\/a><span class=\"os-divider\">. <\/span>Consider [latex]\\frac{x^k+1}{x+1}[\/latex] for [latex]k=1, 3, 5.[\/latex] What do you expect the result to be if [latex]k=7?[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135241323\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135241324\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>Consider [latex]\\frac{x^4-k^4}{x-k}[\/latex] for [latex]k=1, 2, 3.[\/latex] What do you expect the result to be if [latex]k=4?[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132939184\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132939185\" 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-id1165132939184-solution\">57<\/a><span class=\"os-divider\">. <\/span>Consider [latex]\\frac{x^k}{x+1}[\/latex] with [latex]k=1, 2, 3.[\/latex] What do you expect the result to be if [latex]k=4?[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134261659\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134261660\" data-type=\"problem\">\n<p><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>Consider [latex]\\frac{x^k}{x-1}[\/latex] with [latex]k=1, 2, 3.[\/latex] What do you expect the result to be if [latex]k=4?[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135191971\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<p id=\"fs-id1165135191976\">For the following exercises, use synthetic division to determine the quotient involving a complex number.<\/p>\n<div id=\"fs-id1165135191981\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135191982\" 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-id1165135191981-solution\">59<\/a><span class=\"os-divider\">. <\/span> [latex]\\frac{x+1}{x-i}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165135593133\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135593134\" data-type=\"problem\">\n<p><span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span> [latex]\\frac{x^2+1}{x-i}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135593178\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135593179\" 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-id1165135593178-solution\">61<\/a><span class=\"os-divider\">. <\/span> [latex]\\frac{x+1}{x+i}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137777664\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137777665\" data-type=\"problem\">\n<p><span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span> [latex]\\frac{x^2+1}{x+i}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134248021\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134248022\" 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-id1165134248021-solution\">63<\/a><span class=\"os-divider\">. <\/span> [latex]\\frac{x^3+1}{x-i}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165135180069\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<p id=\"fs-id1165135180074\">For the following exercises, use the given length and area of a rectangle to express the width algebraically.<\/p>\n<div id=\"fs-id1165135180078\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135440138\" data-type=\"problem\">\n<p><span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>Length is [latex]x+5,[\/latex] area is [latex]2x^2+9x-5.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135440195\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135440196\" 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-id1165135440195-solution\">65<\/a><span class=\"os-divider\">. <\/span>Length is [latex]2x+5,[\/latex] area is [latex]4x^3+10x^2+6x+15.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135264719\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135264720\" data-type=\"problem\">\n<p><span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>Length is [latex]3x-4,[\/latex] area is [latex]6x^4-8x^3+9x^2-9x-4.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135339554\">For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.<\/p>\n<div id=\"fs-id1165135339559\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135339560\" 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-id1165135339559-solution\">67<\/a><span class=\"os-divider\">. <\/span>Volume is [latex]12x^3+20x^2-21x-36,[\/latex] length is [latex]2x+3,[\/latex] width is [latex]3x-4.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134071557\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134071558\" data-type=\"problem\">\n<p><span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span>Volume is [latex]18x^3-21x^2-40x+48,[\/latex] length is [latex]3x-4,[\/latex] width is [latex]3x-4.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132920308\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132920309\" 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-id1165132920308-solution\">69<\/a><span class=\"os-divider\">. <\/span>Volume is [latex]10x^3+27x^2+2x-24,[\/latex] length is [latex]5x-4,[\/latex] width is [latex]2x+3.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135564178\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135564179\" data-type=\"problem\">\n<p><span class=\"os-number\">70<\/span><span class=\"os-divider\">. <\/span>Volume is [latex]10x^3+30x^2-8x-24,[\/latex] length is [latex]2,[\/latex] width is [latex]x+3.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135514530\">For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.<\/p>\n<div id=\"fs-id1165135514534\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135514535\" 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-id1165135514534-solution\">71<\/a><span class=\"os-divider\">. <\/span>Volume is [latex]\\pi 925x^3-65x^2-29x-3),[\/latex] radius is [latex]5x+1.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134031328\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134031329\" data-type=\"problem\">\n<p><span class=\"os-number\">72. <\/span>Volume is [latex]\\pi (4x^3+12x^2-15x-50),[\/latex] radius is [latex]2x+5.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134258601\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134258602\" 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-id1165134258601-solution\">73<\/a><span class=\"os-divider\">. <\/span>Volume is [latex]\\pi (3x^4+24x^3+46x^2-16x-32),[\/latex] radius is [latex]x+4.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-267-1\">National Park Service. \"Lincoln Memorial Building Statistics.\" http:\/\/www.nps.gov\/linc\/historyculture\/lincoln-memorial-building-statistics.htm. Accessed 4\/3\/2014 <a href=\"#return-footnote-267-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","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-267","chapter","type-chapter","status-publish","hentry"],"part":158,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/267","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":19,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/267\/revisions"}],"predecessor-version":[{"id":1549,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/267\/revisions\/1549"}],"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\/267\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=267"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=267"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=267"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=267"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}