{"id":87,"date":"2025-04-09T17:05:38","date_gmt":"2025-04-09T17:05:38","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/1-4-polynomials-college-algebra-2e-openstax\/"},"modified":"2025-09-22T21:26:32","modified_gmt":"2025-09-22T21:26:32","slug":"1-4-polynomials","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/1-4-polynomials\/","title":{"raw":"1.4 Polynomials","rendered":"1.4 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_f7978ad8-ed27-4fe4-8a19-26a031ba97ad\" 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>Identify the degree and leading coefficient of polynomials.<\/li>\r\n \t<li>Add and subtract polynomials.<\/li>\r\n \t<li>Multiply polynomials.<\/li>\r\n \t<li>Use FOIL to multiply binomials.<\/li>\r\n \t<li>Perform operations with polynomials of several variables.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167339150823\">Maahi is building a little free library (a small house-shaped book repository), whose front is in the shape of a square topped with a triangle. There will be a rectangular door through which people can take and donate books. Maahi wants to find the area of the front of the library so that they can purchase the correct amount of paint. Using the measurements of the front of the house, shown in Figure 1, we can create an expression that combines several variable terms, allowing us to solve this problem and others like it.<\/p>\r\n\r\n\r\n[caption id=\"attachment_1447\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1447\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4-fig-1-300x238.webp\" alt=\"\" width=\"300\" height=\"238\" \/> Figure 1[\/caption]\r\n<p id=\"fs-id1167339220668\">First find the area of the square in square feet.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}A &amp;=&amp; s^2 \\\\&amp;=&amp; (2x)^2 \\\\&amp;=&amp; 4x^2\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1167339240684\">Then find the area of the triangle in square feet.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}A &amp;=&amp; \\frac{1}{2}bh \\\\&amp;=&amp; \\frac{1}{2}(2x)(\\frac{3}{2}) \\\\&amp;=&amp; \\frac{3}{2}x\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1167339173565\">Next find the area of the rectangular door in square feet.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}A &amp;=&amp; lw \\\\&amp;=&amp; x\\cdot 1 \\\\&amp;=&amp; x\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1167339157230\">The area of the front of the library can be found by adding the areas of the square and the triangle, and then subtracting the area of the rectangle. When we do this, we get [latex] 4x^2 + \\frac{3}{2}x - x \\, \\text{ft}^2,[\/latex] or [latex] 4x^2 + \\frac{1}{2}x \\, \\text{ft}^2. [\/latex]<\/p>\r\n<p id=\"fs-id1167339429031\">In this section, we will examine expressions such as this one, which combine several variable terms.<\/p>\r\n\r\n<section id=\"fs-id1167339226748\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Identifying the Degree and Leading Coefficient of Polynomials<\/h2>\r\n<p id=\"fs-id1167339185162\">The formula just found is an example of a <strong><span id=\"term-00003\" data-type=\"term\">polynomial<\/span><\/strong>, which is a sum of or difference of terms, each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as [latex] 384\\pi [\/latex] is known as a <strong><span id=\"term-00004\" data-type=\"term\">coefficient<\/span><\/strong>. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product [latex] a_ix^i, [\/latex] such as [latex] 384\\pi w [\/latex] is a <strong><span id=\"term-00005\" data-type=\"term\">term of a polynomial<\/span><\/strong>. If a term does not contain a variable, it is called a <em data-effect=\"italics\">constant<\/em>.<\/p>\r\n<p id=\"fs-id1167339138167\">A polynomial containing only one term, such as [latex] 5x^4, [\/latex] is called a <strong><span id=\"term-00006\" data-type=\"term\">monomial<\/span><\/strong>. A polynomial containing two terms, such as [latex] 2x-9, [\/latex] is called a <strong><span id=\"term-00007\" data-type=\"term\">binomial<\/span><\/strong>. A polynomial containing three terms, such as [latex] -3x^2+8x-7, [\/latex] is called a <strong><span id=\"term-00008\" data-type=\"term\">trinomial<\/span><\/strong>.<\/p>\r\n<p id=\"fs-id1167339222212\">We can find the <strong><span id=\"term-00009\" data-type=\"term\">degree<\/span> <\/strong>of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the<strong> <span id=\"term-00010\" data-type=\"term\">leading term<\/span><\/strong> because it is usually written first. The coefficient of the leading term is called the<strong> <span id=\"term-00011\" data-type=\"term\">leading coefficient<\/span><\/strong>. When a polynomial is written so that the powers are descending, we say that it is in standard form.<span id=\"fs-id1167339243309\" data-type=\"media\" data-alt=\"A polynomial reading: a sub n times x to the nth power plus and so on plus a sub 2 times x squared plus a sub one times x plus a subzero is shown. The a in the term a sub n is labeled: leading coefficient. The n in the term x to the nth power is labeled: degree. Finally, the entire term is labeled as: Leading term.\"><\/span><\/p>\r\n<img class=\"size-medium wp-image-1448 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4-300x73.webp\" alt=\"\" width=\"300\" height=\"73\" \/>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Polynomials<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1167339155557\">A <strong>polynomial<\/strong> is an expression that can be written in the form<\/p>\r\n<p style=\"text-align: center;\">[latex] a_n x^n + ... + a_2 x^2 + a_1 x + a_0 [\/latex]<\/p>\r\n<p id=\"fs-id1167339220598\">Each real number [latex] a_i [\/latex] is called a <strong>coefficient<\/strong>. The number [latex] a_0 [\/latex] that is not multiplied by a variable is called a <em data-effect=\"italics\">constant<\/em>. Each product [latex] a_ix^i [\/latex] is a<strong> term of a polynomial.<\/strong> The highest power of the variable that occurs in the polynomial is called the <strong>degree<\/strong> of a polynomial. The <strong>leading term<\/strong> is the term with the highest power, and its coefficient is called the <strong>leading coefficient<\/strong>.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/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<p id=\"fs-id1167339212575\"><strong>Given a polynomial expression, identify the degree and leading coefficient<\/strong>.<\/p>\r\n\r\n<ol id=\"fs-id1167339329242\" type=\"1\">\r\n \t<li>Find the highest power of <em data-effect=\"italics\">x<\/em> to determine the degree.<\/li>\r\n \t<li>Identify the term containing the highest power of <em data-effect=\"italics\">x<\/em> to find the leading term.<\/li>\r\n \t<li>Identify the coefficient of the leading term.<\/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: Identifying the Degree and Leading Coefficient of a Polynomial<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFor the following polynomials, identify the degree, the leading term, and the leading coefficient.\r\n\r\n(a) [latex] 3+2x^2-4x^3 [\/latex]\r\n\r\n(b) [latex] 5t^5-2t^3+7t [\/latex]\r\n\r\n(c) [latex] 6p-p^3-2 [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a) The highest power of <em data-effect=\"italics\">x<\/em> is [latex] 3, [\/latex] so the degree is [latex] 3. [\/latex] The leading term is the term containing that degree, [latex] -4x^3. [\/latex]\u00a0 The leading coefficient is the coefficient of that term, [latex] -4. [\/latex]\r\n\r\n(b) The highest power of <em data-effect=\"italics\">t<\/em> is [latex] 5, [\/latex] so the degree is [latex] 5. [\/latex] The leading term is the term containing that degree [latex] 5t^5. [\/latex] The leading coefficient is the coefficient of that term, [latex] 5. [\/latex]\r\n\r\n(c) The highest power of <em data-effect=\"italics\">p<\/em> is [latex] 3, [\/latex] so the degree is [latex] 3. [\/latex] The leading term is the term containing that degree, [latex] -p^3. [\/latex] The leading coefficient is the coefficient of that term, [latex] -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 #1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIdentify the degree, leading term, and leading coefficient of the polynomial [latex] 4x^2-x^6+2x-6. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1167339243781\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Adding and Subtracting Polynomials<\/h2>\r\n<p id=\"fs-id1167339223237\">We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example, [latex] 5x^2 [\/latex] and [latex] -2x^2 [\/latex] are like terms, and can be added to get [latex] 3x^2, [\/latex] but [latex] 3x [\/latex] and [latex] 3x^2 [\/latex] are not like terms, and therefore cannot be added.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1167339199765\"><strong>Given multiple polynomials, add or subtract them to simplify the expressions.\r\n<\/strong><\/p>\r\n\r\n<ol type=\"1\">\r\n \t<li>Combine like terms.<\/li>\r\n \t<li>Simplify and write in standard form.<\/li>\r\n<\/ol>\r\n<ol id=\"fs-id1167339199769\" type=\"1\"><\/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 2: Adding Polynomials<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the sum.\r\n\r\n[latex] (12x^2+9x-21)+(4x^3+8x^2-5x+20) [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>[latex] \\begin{array}{ll}4x^3 + (12x^2 + 8x^2) + (9x - 5x) + (-21 + 20) \\, &amp; \\quad \\text{ Combine like terms.} \\\\4x^3 + 20x^2 + 4x - 1 &amp; \\quad \\text{ Simplify.}\\end{array} [\/latex]\r\n<h3>Analysis<\/h3>\r\nWe can check our answers to these types of problems using a graphing calculator. To check, graph the problem as given along with the simplified answer. The two graphs should be equivalent. Be sure to use the same window to compare the graphs. Using different windows can make the expressions seem equivalent when they are not.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\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\nFind the sum.\r\n\r\n[latex] (2x^3+5x^2-x+1)+(2x^2-3x-4) [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3: Subtracting Polynomials<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the difference.\r\n\r\n[latex] (7x^4-x^2+6x+1)-(5x^3-2x^2+3x+2) [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>[latex] \\begin{array}{ll}7x^4 - x^2 + 6x + 1 - 5x^3 + 2x^2 - 3x - 2 &amp; \\quad \\text{Distribute negative sign.} \\\\7x^4 - 5x^3 + x^2 + 6x - 3x + 1 - 2 &amp; \\quad \\text{Group like terms.} \\\\7x^4 - 5x^3 + x^2 + 3x - 1 &amp; \\quad \\text{Combine\/simplify.}\\end{array} [\/latex]\r\n<h3>Analysis<\/h3>\r\nNote that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.\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\nFind the difference.\r\n\r\n[latex] (-7x^3-7x^2+6x-2)-(4x^3-6x^2-x+7) [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1167339214258\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Multiplying Polynomials<\/h2>\r\n<p id=\"fs-id1167339139443\">Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the <strong><span id=\"term-00018\" data-type=\"term\">FOIL<\/span><\/strong> method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.<\/p>\r\n\r\n<section id=\"fs-id1167339198311\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Multiplying Polynomials Using the Distributive Property<\/h3>\r\n<p id=\"fs-id1167339198316\">To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of the polynomial. We can distribute the [latex] 2 [\/latex] in [latex] 2(x+7) [\/latex] to obtain the equivalent expression [latex] 2x+14. [\/latex] When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1167339170371\"><strong>Given the multiplication of two polynomials, use the distributive property to simplify the expression.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1167339170375\" type=\"1\">\r\n \t<li>Multiply each term of the first polynomial by each term of the second.<\/li>\r\n \t<li>Combine like terms.<\/li>\r\n \t<li>Simplify.<\/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 4: Multiplying Polynomials Using the Distributive Property<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the product.\r\n\r\n[latex] (2x+1)(3x^2-x+4) [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>[latex] \\begin{array}{ll}2x(3x^2-x+4)+1(3x^2-x+4) &amp; \\quad \\text{Use the distributive property.} \\\\(6x^3-2x^2+8x)+(3x^2-x+4) &amp; \\quad \\text{Multiply.} \\\\6x^3+(-2x^2+3x^2)+(8x-x)+4 &amp; \\quad \\text{Combine like terms.} \\\\6x^3+x^2+7x+4 &amp; \\quad \\text{Simplify.}\\end{array} [\/latex]\r\n<h3>Analysis<\/h3>\r\nWe can use a table to keep track of our work, as shown in Table 1. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 1<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 25%;\"><\/td>\r\n<td style=\"width: 25%;\">[latex] 3x^2 [\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex] -x [\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex] +4 [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex] 2x [\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex] 6x^3 [\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex] -2x^2 [\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex] 8x [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex] +1 [\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex] 3x^2 [\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex] -x [\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex] 4 [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section><\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the product.\r\n\r\n[latex] (3x+2)(x^3-4x^2+7) [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1167339243382\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Using FOIL to Multiply Binomials<\/h3>\r\n<p id=\"fs-id1167339243388\">A shortcut called FOIL is sometimes used to find the product of two binomials. It is called FOIL because we multiply the <strong>f<\/strong>irst terms, the <strong>o<\/strong>uter terms, the <strong>i<\/strong>nner terms, and then the <strong>l<\/strong>ast terms of each binomial.<span id=\"fs-id1167339168201\" data-type=\"media\" data-alt=\"Two quantities in parentheses are being multiplied, the first being: a times x plus b and the second being: c times x plus d. This expression equals ac times x squared plus ad times x plus bc times x plus bd. The terms ax and cx are labeled: First Terms. The terms ax and d are labeled: Outer Terms. The terms b and cx are labeled: Inner Terms. The terms b and d are labeled: Last Terms.\"><\/span><\/p>\r\n<img class=\"size-medium wp-image-1449 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.1-300x87.webp\" alt=\"\" width=\"300\" height=\"87\" \/>\r\n<p id=\"fs-id1167339429089\">The FOIL method arises out of the distributive property. We are simply multiplying each term of the first binomial by each term of the second binomial, and then combining like terms.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1167339240108\"><strong>Given two binomials, use FOIL to simplify the expression.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1167339240112\" type=\"1\">\r\n \t<li>Multiply the first terms of each binomial.<\/li>\r\n \t<li>Multiply the outer terms of the binomials.<\/li>\r\n \t<li>Multiply the inner terms of the binomials.<\/li>\r\n \t<li>Multiply the last terms of each binomial.<\/li>\r\n \t<li>Add the products.<\/li>\r\n \t<li>Combine like terms and simplify.<\/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 5: Using FOIL to Multiply Binomials<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse FOIL to find the product.\r\n\r\n[latex] (2x-18)(3x+3) [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Find the product of the first terms.\r\n\r\n<img class=\"alignnone size-medium wp-image-1450\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5-300x32.webp\" alt=\"\" width=\"300\" height=\"32\" \/>\r\n\r\nFind the product of the outer terms.\r\n\r\n<img class=\"alignnone size-medium wp-image-1451\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.1-300x32.webp\" alt=\"\" width=\"300\" height=\"32\" \/>\r\n\r\nFind the product of the inner terms.\r\n\r\n<img class=\"alignnone size-medium wp-image-1452\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.2-300x32.webp\" alt=\"\" width=\"300\" height=\"32\" \/>\r\n\r\nFind the product of the last terms.\r\n\r\n<img class=\"alignnone size-medium wp-image-1453\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.3-300x32.webp\" alt=\"\" width=\"300\" height=\"32\" \/>\r\n\r\n[latex] \\begin{array}{ll}6x^2+6x-54x-54 &amp; \\quad \\text{Add the products.} \\\\6x^2+(6x-54x)-54 &amp; \\quad \\text{Combine like terms.} \\\\6x^2-48x-54 &amp; \\quad \\text{Simplify.} \\end{array} [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse FOIL to find the product.\r\n\r\n[latex] (x+7)(3x-5) [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1167339428910\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Perfect Square Trinomials<\/h3>\r\n<p id=\"fs-id1167339432924\">Certain binomial products have special forms. When a binomial is squared, the result is called a <span id=\"term-00019\" data-type=\"term\"><strong>perfect square trinomia<\/strong>l<\/span>. We can find the square by multiplying the binomial by itself. However, there is a special form that each of these perfect square trinomials takes, and memorizing the form makes squaring binomials much easier and faster. Let\u2019s look at a few perfect square trinomials to familiarize ourselves with the form.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}\\, (x + 5)^2 &amp;=&amp; x^2 + 10x + 25 \\\\(x - 3)^2 &amp;=&amp; \\, x^2 - 6x + 9 \\\\(4x - 1)^2 &amp;=&amp; 16x^2 - 8x + 1\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1167339224293\">Notice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of each trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms. Lastly, we see that the first sign of the trinomial is the same as the sign of the binomial.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Perfect Square Trinomials<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhen a binomial is squared, the result is the first term squared added to double the product of both terms and the last term squared.\r\n<p style=\"text-align: center;\">[latex] (x+a)^2=(x+a)(x+a)=x^2+2ax+a^2 [\/latex]<\/p>\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<p id=\"fs-id1167339220533\"><strong>Given a binomial, square it using the formula for perfect square trinomials.\r\n<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1167339231805\" type=\"1\">\r\n \t<li>Square the first term of the binomial.<\/li>\r\n \t<li>Square the last term of the binomial.<\/li>\r\n \t<li>For the middle term of the trinomial, double the product of the two terms.<\/li>\r\n \t<li>Add and simplify.<\/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 6: Expanding Perfect Squares<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nExpand [latex] (3x-8)^2. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Begin by squaring the first term and the last term. For the middle term of the trinomial, double the product of the two terms.\r\n<p style=\"text-align: center;\">[latex] (3x)^2-2(3x)(8)+(-8)^2 [\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex] 9x^2-48x+64 [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">Expand [latex] (4x-1)^2. [\/latex]<\/div>\r\n<\/div>\r\n<section id=\"fs-id1167339429838\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Difference of Squares<\/h3>\r\n<p id=\"fs-id1167339188447\">Another special product is called the <strong><span id=\"term-00020\" data-type=\"term\">difference of squares<\/span>,<\/strong> which occurs when we multiply a binomial by another binomial with the same terms but the opposite sign. Let\u2019s see what happens when we multiply [latex] (x+1)(x-1) [\/latex] using the FOIL method.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}\\, (x+1)(x-1) &amp;=&amp; x^2-x+x-1 \\\\ &amp;=&amp; \\, x^2-1 \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1167339432898\">The middle term drops out, resulting in a difference of squares. Just as we did with the perfect squares, let\u2019s look at a few examples.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}\\, (x+5)(x-5) &amp;=&amp; x^2-25 \\\\ (x+11)(x-11) &amp;=&amp; \\, x^2-121 \\\\ (2x+3)(2x-3) &amp;=&amp; 4x^2-9 \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1167339220812\">Because the sign changes in the second binomial, the outer and inner terms cancel each other out, and we are left only with the square of the first term minus the square of the last term.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1167339220824\"><strong>Q: Is there a special form for the sum of squares?<\/strong><\/p>\r\n<p id=\"fs-id1167339199657\"><em data-effect=\"italics\">A: No. The difference of squares occurs because the opposite signs of the binomials cause the middle terms to disappear. There are no two binomials that multiply to equal a sum of squares.<\/em><\/p>\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\">Difference of Squares<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhen a binomial is multiplied by a binomial with the same terms separated by the opposite sign, the result is the square of the first term minus the square of the last term.\r\n<p style=\"text-align: center;\">[latex] (a+b)(a-b)=a^2-b^2 [\/latex]<\/p>\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<p id=\"fs-id1167339273978\"><strong>Given a binomial multiplied by a binomial with the same terms but the opposite sign, find the difference of squares.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1167339273983\" type=\"1\">\r\n \t<li>Square the first term of the binomials.<\/li>\r\n \t<li>Square the last term of the binomials.<\/li>\r\n \t<li>Subtract the square of the last term from the square of the first term.<\/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 7: Multiplying Binomials Resulting in a Difference of Squares<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nMultiply [latex] (9x+4)(9x-4). [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Square the first term to get [latex] (9x)^2=81x^2. [\/latex] Square the last term to get [latex] 4^2=16. [\/latex] Subtract the square of the last term from the square of the first term to find the product of [latex] 81x^2-16. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nMultiply [latex] (2x+7)(2x-7). [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1167339138125\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Performing Operations with Polynomials of Several Variables<\/h2>\r\n<p id=\"fs-id1167339138130\">We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll}(a + 2b)(4a - b - c) &amp; \\\\a(4a - b - c) + 2b(4a - b - c) &amp; \\quad \\text{Use the distributive property.} \\\\4a^2 - ab - ac + 8ab - 2b^2 - 2bc &amp; \\quad \\text{Multiply.} \\\\4a^2 + (-ab + 8ab) - ac - 2b^2 - 2bc &amp; \\quad \\text{Combine like terms.} \\\\4a^2 + 7ab - ac - 2bc - 2b^2 &amp; \\quad \\text{Simplify.}\\end{array} [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 8: Multiplying Polynomials Containing Several Variables<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nMultiply [latex] (x+4)(3x-2y+5). [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Follow the same steps that we used to multiply polynomials containing only one variable.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll}x(3x-2y+5)+4_3x-2y+5) &amp; \\quad \\text{Use the distributive property.} \\\\3x^2-2xy+5x+12x-8y+20 &amp; \\quad \\text{Multiply.} \\\\3x^2-2xy+(5x+12x)-8y+20 &amp; \\quad \\text{Combine like terms.} \\\\3x^2-2xy+17x-8y+20 &amp; \\quad \\text{Simplify.}\\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nMultiply [latex] (3x-1)(2x+7y-9). [\/latex]\r\n\r\n<\/div>\r\n<\/div>\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 polynomials.\r\n<ul>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/addsubpoly\" target=\"_blank\" rel=\"noopener nofollow\">Adding and Subtracting Polynomials<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/multiplpoly\" target=\"_blank\" rel=\"noopener nofollow\">Multiplying Polynomials<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/specialpolyprod\" target=\"_blank\" rel=\"noopener nofollow\">Special Products of Polynomials<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\r\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">1.4 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1167339184236\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1167339184243\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1167339184249\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339184250\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339184249-solution\">1<\/a><span class=\"os-divider\">. <\/span>Evaluate the following statement: The degree of a polynomial in standard form is the exponent of the leading term. Explain why the statement is true or false.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339216138\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339216140\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>Many times, multiplying two binomials with two variables results in a trinomial. This is not the case when there is a difference of two squares. Explain why the product in this case is also a binomial.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339216146\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339216147\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339216146-solution\">3<\/a><span class=\"os-divider\">. <\/span>You can multiply polynomials with any number of terms and any number of variables using four basic steps over and over until you reach the expanded polynomial. What are the four steps?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339216157\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339216158\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>State whether the following statement is true and explain why or why not: A trinomial is always a higher degree than a monomial.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1167339216165\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id1167339216170\">For the following exercises, identify the degree of the polynomial.<\/p>\r\n\r\n<div id=\"fs-id1167339216173\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339216174\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339216173-solution\">5<\/a><span class=\"os-divider\">. <\/span> [latex] 7x-2x^2+13 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339299774\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339299775\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex] 14m^3+m^2-16m+8 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339299820\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339299821\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339299820-solution\">7<\/a><span class=\"os-divider\">.<\/span> [latex] -625a^8+16b^4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339268846\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339268847\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">.<\/span> [latex] 200p-30p^2m+40m^3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339223145\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339223146\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339223145-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex] x^2+4x+4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339223181\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339223182\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex] 6y^4-y^5+3y-4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1167339432583\">For the following exercises, find the sum or difference.<\/p>\r\n\r\n<div id=\"fs-id1167339432587\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339432588\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339432587-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex] (12x^2+3x)-(8x^2-19) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339185745\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339185746\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex] (4z^3+8z^2-z)+(-2z^2+z+6) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339306370\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339306371\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339306370-solution\">13<\/a><span class=\"os-divider\">.<\/span> [latex] (6w^2+24w+24)-(3w^2-6w+3) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339223833\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339223834\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex] (7a^3+6a^2-4a-13)+(_3a^3-4a^2+6a+17) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339262640\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339262641\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339262640-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex] (11b^4-6b^3+18b^2-4b+8)-(3b^3+6b^2+3b) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339196660\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339196661\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex]- (49p^2-25)+(16p^4-32p^2+16) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1167339196741\">For the following exercises, find the product.<\/p>\r\n\r\n<div id=\"fs-id1167339196745\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339196746\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339196745-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] (4x+2)(6x-4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339259698\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339259699\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] (14c^2+4c)(2c^2-3c) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339223391\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339223392\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339223391-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex] (6b^2-6)(4b^2-4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339223497\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339223498\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex] (3d-5)(2d+9) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339306553\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339306554\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339306553-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex] (9v-11)(11v-9) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339286443\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339286444\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex] (4t^2+7t)(-3t^2+4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339286513\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339286514\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339286513-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] (8n-4)(n^2+9) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1167339199464\">For the following exercises, expand the binomial.<\/p>\r\n\r\n<div id=\"fs-id1167339199468\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339199469\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">.<\/span> [latex] (4x+5)^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339199508\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339199509\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339199508-solution\">25<\/a><span class=\"os-divider\">.<\/span> [latex] (3y-7)^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339220230\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339220232\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">.<\/span> [latex] (12-4x)^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339220271\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339220272\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339220271-solution\">27<\/a><span class=\"os-divider\">.<\/span> [latex] (4p+9)^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339220345\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339220346\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">.<\/span> [latex] (2m-3)^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339226325\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339226326\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339226325-solution\">29<\/a><span class=\"os-divider\">.<\/span> [latex] (3y-6)^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339226399\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339226400\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">.<\/span> [latex] (9b+1)^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1167339196445\">For the following exercises, multiply the binomials.<\/p>\r\n\r\n<div id=\"fs-id1167339196448\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339196449\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339196448-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex] (4c+1)(4c-1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339196525\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339196526\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex] (9a-4)(9a+4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339196575\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339196576\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339196575-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex] (15n-6)(15n+6) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339260474\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339260475\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex] (25b+20(25b-2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339260524\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339260525\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339260524-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex] (4+4m)(4-4m) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339239069\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339239070\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex] (14p+7)(14p-7) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339239119\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339239120\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339239119-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex] (11q-10)(11q+10) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1167339321358\">For the following exercises, multiply the polynomials.<\/p>\r\n\r\n<div id=\"fs-id1167339321361\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339321362\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex] (2x^2+2x+1)(4x-1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339321425\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339321426\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339321425-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex] (4t^2+t-7)(4t^2-1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339138836\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339138837\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex] (x-1)(x^2-2x+1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339138896\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339138897\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339138896-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex] (y-2)(y^2-4y-9) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339273830\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339273831\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex] (6k-5)(6k^2+5k-1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339273894\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339273895\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339273894-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex] (3p^2+2p-10)(p-1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339344452\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339344453\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex] (4m-13)(2m^2-7m+9) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339344516\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339344517\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339344516-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex] (a+b)(a-b) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339213948\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339213949\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex] (4x-6y)(6x-4y) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339214002\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339214003\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339214002-solution\">47<\/a><span class=\"os-divider\">.<\/span> [latex] (4t-5u)^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339214089\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339214090\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex] (9m+4n-1)(2m+8) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339222871\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339222872\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339222871-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex] (4t-x)(t-x+1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339222977\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339222978\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50<\/span><span class=\"os-divider\">.<\/span> [latex] (b^2-1)(a^2+2ab+b^2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339223053\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339223054\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339223053-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex] (4r-d)(6r+7d) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339225658\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339225659\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">.<\/span> [latex] (x+y)(x^2-xy+y^2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1167339225725\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<div id=\"fs-id1167339225730\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339225731\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339225730-solution\">53<\/a><span class=\"os-divider\">. <\/span>A developer wants to purchase a plot of land to build a house. The area of the plot can be described by the following expression: [latex] (4x+1)(8x-3) [\/latex] where <em data-effect=\"italics\">x<\/em> is measured in meters. Multiply the binomials to find the area of the plot in standard form.\r\n<div class=\"os-problem-container\"><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339242981\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339242982\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>A prospective buyer wants to know how much grain a specific silo can hold. The area of the floor of the silo is [latex] (2x+9)^2. [\/latex] The height of the silo is [latex] 10x+10, [\/latex] where <em data-effect=\"italics\">x<\/em> is measured in feet. Expand the square and multiply by the height to find the expression that shows how much grain the silo can hold.\r\n<div class=\"os-problem-container\"><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1167339243059\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<p id=\"fs-id1167339243064\">For the following exercises, perform the given operations.<\/p>\r\n\r\n<div id=\"fs-id1167339243067\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339243068\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339243067-solution\">55<\/a><span class=\"os-divider\">. <\/span> [latex] (4t-7)^2(2t+1)-(4t^2+2t+11) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339230814\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339230815\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span> [latex] (3b+6)(3b-6)(9b^2-36) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339230891\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1167339230892\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339230891-solution\">57<\/a><span class=\"os-divider\">. <\/span> [latex] (a^2+4ac+4c^2)(a^2-4c^2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_f7978ad8-ed27-4fe4-8a19-26a031ba97ad\" 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>Identify the degree and leading coefficient of polynomials.<\/li>\n<li>Add and subtract polynomials.<\/li>\n<li>Multiply polynomials.<\/li>\n<li>Use FOIL to multiply binomials.<\/li>\n<li>Perform operations with polynomials of several variables.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339150823\">Maahi is building a little free library (a small house-shaped book repository), whose front is in the shape of a square topped with a triangle. There will be a rectangular door through which people can take and donate books. Maahi wants to find the area of the front of the library so that they can purchase the correct amount of paint. Using the measurements of the front of the house, shown in Figure 1, we can create an expression that combines several variable terms, allowing us to solve this problem and others like it.<\/p>\n<figure id=\"attachment_1447\" aria-describedby=\"caption-attachment-1447\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1447\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4-fig-1-300x238.webp\" alt=\"\" width=\"300\" height=\"238\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4-fig-1-300x238.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4-fig-1-65x51.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4-fig-1-225x178.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4-fig-1-350x277.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4-fig-1.webp 360w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1447\" class=\"wp-caption-text\">Figure 1<\/figcaption><\/figure>\n<p id=\"fs-id1167339220668\">First find the area of the square in square feet.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}A &=& s^2 \\\\&=& (2x)^2 \\\\&=& 4x^2\\end{array}[\/latex]<\/p>\n<p id=\"fs-id1167339240684\">Then find the area of the triangle in square feet.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}A &=& \\frac{1}{2}bh \\\\&=& \\frac{1}{2}(2x)(\\frac{3}{2}) \\\\&=& \\frac{3}{2}x\\end{array}[\/latex]<\/p>\n<p id=\"fs-id1167339173565\">Next find the area of the rectangular door in square feet.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}A &=& lw \\\\&=& x\\cdot 1 \\\\&=& x\\end{array}[\/latex]<\/p>\n<p id=\"fs-id1167339157230\">The area of the front of the library can be found by adding the areas of the square and the triangle, and then subtracting the area of the rectangle. When we do this, we get [latex]4x^2 + \\frac{3}{2}x - x \\, \\text{ft}^2,[\/latex] or [latex]4x^2 + \\frac{1}{2}x \\, \\text{ft}^2.[\/latex]<\/p>\n<p id=\"fs-id1167339429031\">In this section, we will examine expressions such as this one, which combine several variable terms.<\/p>\n<section id=\"fs-id1167339226748\" data-depth=\"1\">\n<h2 data-type=\"title\">Identifying the Degree and Leading Coefficient of Polynomials<\/h2>\n<p id=\"fs-id1167339185162\">The formula just found is an example of a <strong><span id=\"term-00003\" data-type=\"term\">polynomial<\/span><\/strong>, which is a sum of or difference of terms, each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as [latex]384\\pi[\/latex] is known as a <strong><span id=\"term-00004\" data-type=\"term\">coefficient<\/span><\/strong>. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product [latex]a_ix^i,[\/latex] such as [latex]384\\pi w[\/latex] is a <strong><span id=\"term-00005\" data-type=\"term\">term of a polynomial<\/span><\/strong>. If a term does not contain a variable, it is called a <em data-effect=\"italics\">constant<\/em>.<\/p>\n<p id=\"fs-id1167339138167\">A polynomial containing only one term, such as [latex]5x^4,[\/latex] is called a <strong><span id=\"term-00006\" data-type=\"term\">monomial<\/span><\/strong>. A polynomial containing two terms, such as [latex]2x-9,[\/latex] is called a <strong><span id=\"term-00007\" data-type=\"term\">binomial<\/span><\/strong>. A polynomial containing three terms, such as [latex]-3x^2+8x-7,[\/latex] is called a <strong><span id=\"term-00008\" data-type=\"term\">trinomial<\/span><\/strong>.<\/p>\n<p id=\"fs-id1167339222212\">We can find the <strong><span id=\"term-00009\" data-type=\"term\">degree<\/span> <\/strong>of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the<strong> <span id=\"term-00010\" data-type=\"term\">leading term<\/span><\/strong> because it is usually written first. The coefficient of the leading term is called the<strong> <span id=\"term-00011\" data-type=\"term\">leading coefficient<\/span><\/strong>. When a polynomial is written so that the powers are descending, we say that it is in standard form.<span id=\"fs-id1167339243309\" data-type=\"media\" data-alt=\"A polynomial reading: a sub n times x to the nth power plus and so on plus a sub 2 times x squared plus a sub one times x plus a subzero is shown. The a in the term a sub n is labeled: leading coefficient. The n in the term x to the nth power is labeled: degree. Finally, the entire term is labeled as: Leading term.\"><\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1448 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4-300x73.webp\" alt=\"\" width=\"300\" height=\"73\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4-300x73.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4-65x16.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4-225x55.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4-350x86.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.webp 560w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Polynomials<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1167339155557\">A <strong>polynomial<\/strong> is an expression that can be written in the form<\/p>\n<p style=\"text-align: center;\">[latex]a_n x^n + ... + a_2 x^2 + a_1 x + a_0[\/latex]<\/p>\n<p id=\"fs-id1167339220598\">Each real number [latex]a_i[\/latex] is called a <strong>coefficient<\/strong>. The number [latex]a_0[\/latex] that is not multiplied by a variable is called a <em data-effect=\"italics\">constant<\/em>. Each product [latex]a_ix^i[\/latex] is a<strong> term of a polynomial.<\/strong> The highest power of the variable that occurs in the polynomial is called the <strong>degree<\/strong> of a polynomial. The <strong>leading term<\/strong> is the term with the highest power, and its coefficient is called the <strong>leading coefficient<\/strong>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1167339212575\"><strong>Given a polynomial expression, identify the degree and leading coefficient<\/strong>.<\/p>\n<ol id=\"fs-id1167339329242\" type=\"1\">\n<li>Find the highest power of <em data-effect=\"italics\">x<\/em> to determine the degree.<\/li>\n<li>Identify the term containing the highest power of <em data-effect=\"italics\">x<\/em> to find the leading term.<\/li>\n<li>Identify the coefficient of the leading term.<\/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: Identifying the Degree and Leading Coefficient of a Polynomial<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>For the following polynomials, identify the degree, the leading term, and the leading coefficient.<\/p>\n<p>(a) [latex]3+2x^2-4x^3[\/latex]<\/p>\n<p>(b) [latex]5t^5-2t^3+7t[\/latex]<\/p>\n<p>(c) [latex]6p-p^3-2[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a) The highest power of <em data-effect=\"italics\">x<\/em> is [latex]3,[\/latex] so the degree is [latex]3.[\/latex] The leading term is the term containing that degree, [latex]-4x^3.[\/latex]\u00a0 The leading coefficient is the coefficient of that term, [latex]-4.[\/latex]<\/p>\n<p>(b) The highest power of <em data-effect=\"italics\">t<\/em> is [latex]5,[\/latex] so the degree is [latex]5.[\/latex] The leading term is the term containing that degree [latex]5t^5.[\/latex] The leading coefficient is the coefficient of that term, [latex]5.[\/latex]<\/p>\n<p>(c) The highest power of <em data-effect=\"italics\">p<\/em> is [latex]3,[\/latex] so the degree is [latex]3.[\/latex] The leading term is the term containing that degree, [latex]-p^3.[\/latex] The leading coefficient is the coefficient of that term, [latex]-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 #1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Identify the degree, leading term, and leading coefficient of the polynomial [latex]4x^2-x^6+2x-6.[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1167339243781\" data-depth=\"1\">\n<h2 data-type=\"title\">Adding and Subtracting Polynomials<\/h2>\n<p id=\"fs-id1167339223237\">We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example, [latex]5x^2[\/latex] and [latex]-2x^2[\/latex] are like terms, and can be added to get [latex]3x^2,[\/latex] but [latex]3x[\/latex] and [latex]3x^2[\/latex] are not like terms, and therefore cannot be added.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1167339199765\"><strong>Given multiple polynomials, add or subtract them to simplify the expressions.<br \/>\n<\/strong><\/p>\n<ol type=\"1\">\n<li>Combine like terms.<\/li>\n<li>Simplify and write in standard form.<\/li>\n<\/ol>\n<ol id=\"fs-id1167339199769\" type=\"1\"><\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Adding Polynomials<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the sum.<\/p>\n<p>[latex](12x^2+9x-21)+(4x^3+8x^2-5x+20)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>[latex]\\begin{array}{ll}4x^3 + (12x^2 + 8x^2) + (9x - 5x) + (-21 + 20) \\, & \\quad \\text{ Combine like terms.} \\\\4x^3 + 20x^2 + 4x - 1 & \\quad \\text{ Simplify.}\\end{array}[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>We can check our answers to these types of problems using a graphing calculator. To check, graph the problem as given along with the simplified answer. The two graphs should be equivalent. Be sure to use the same window to compare the graphs. Using different windows can make the expressions seem equivalent when they are not.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\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>Find the sum.<\/p>\n<p>[latex](2x^3+5x^2-x+1)+(2x^2-3x-4)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Subtracting Polynomials<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the difference.<\/p>\n<p>[latex](7x^4-x^2+6x+1)-(5x^3-2x^2+3x+2)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>[latex]\\begin{array}{ll}7x^4 - x^2 + 6x + 1 - 5x^3 + 2x^2 - 3x - 2 & \\quad \\text{Distribute negative sign.} \\\\7x^4 - 5x^3 + x^2 + 6x - 3x + 1 - 2 & \\quad \\text{Group like terms.} \\\\7x^4 - 5x^3 + x^2 + 3x - 1 & \\quad \\text{Combine\/simplify.}\\end{array}[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.<\/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>Find the difference.<\/p>\n<p>[latex](-7x^3-7x^2+6x-2)-(4x^3-6x^2-x+7)[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1167339214258\" data-depth=\"1\">\n<h2 data-type=\"title\">Multiplying Polynomials<\/h2>\n<p id=\"fs-id1167339139443\">Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the <strong><span id=\"term-00018\" data-type=\"term\">FOIL<\/span><\/strong> method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.<\/p>\n<section id=\"fs-id1167339198311\" data-depth=\"2\">\n<h3 data-type=\"title\">Multiplying Polynomials Using the Distributive Property<\/h3>\n<p id=\"fs-id1167339198316\">To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of the polynomial. We can distribute the [latex]2[\/latex] in [latex]2(x+7)[\/latex] to obtain the equivalent expression [latex]2x+14.[\/latex] When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1167339170371\"><strong>Given the multiplication of two polynomials, use the distributive property to simplify the expression.<\/strong><\/p>\n<ol id=\"fs-id1167339170375\" type=\"1\">\n<li>Multiply each term of the first polynomial by each term of the second.<\/li>\n<li>Combine like terms.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Multiplying Polynomials Using the Distributive Property<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the product.<\/p>\n<p>[latex](2x+1)(3x^2-x+4)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>[latex]\\begin{array}{ll}2x(3x^2-x+4)+1(3x^2-x+4) & \\quad \\text{Use the distributive property.} \\\\(6x^3-2x^2+8x)+(3x^2-x+4) & \\quad \\text{Multiply.} \\\\6x^3+(-2x^2+3x^2)+(8x-x)+4 & \\quad \\text{Combine like terms.} \\\\6x^3+x^2+7x+4 & \\quad \\text{Simplify.}\\end{array}[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>We can use a table to keep track of our work, as shown in Table 1. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 1<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 25%;\"><\/td>\n<td style=\"width: 25%;\">[latex]3x^2[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]-x[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]+4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">[latex]2x[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]6x^3[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]-2x^2[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]8x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">[latex]+1[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]3x^2[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]-x[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the product.<\/p>\n<p>[latex](3x+2)(x^3-4x^2+7)[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1167339243382\" data-depth=\"2\">\n<h3 data-type=\"title\">Using FOIL to Multiply Binomials<\/h3>\n<p id=\"fs-id1167339243388\">A shortcut called FOIL is sometimes used to find the product of two binomials. It is called FOIL because we multiply the <strong>f<\/strong>irst terms, the <strong>o<\/strong>uter terms, the <strong>i<\/strong>nner terms, and then the <strong>l<\/strong>ast terms of each binomial.<span id=\"fs-id1167339168201\" data-type=\"media\" data-alt=\"Two quantities in parentheses are being multiplied, the first being: a times x plus b and the second being: c times x plus d. This expression equals ac times x squared plus ad times x plus bc times x plus bd. The terms ax and cx are labeled: First Terms. The terms ax and d are labeled: Outer Terms. The terms b and cx are labeled: Inner Terms. The terms b and d are labeled: Last Terms.\"><\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1449 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.1-300x87.webp\" alt=\"\" width=\"300\" height=\"87\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.1-300x87.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.1-65x19.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.1-225x65.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.1-350x101.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.1.webp 601w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"fs-id1167339429089\">The FOIL method arises out of the distributive property. We are simply multiplying each term of the first binomial by each term of the second binomial, and then combining like terms.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1167339240108\"><strong>Given two binomials, use FOIL to simplify the expression.<\/strong><\/p>\n<ol id=\"fs-id1167339240112\" type=\"1\">\n<li>Multiply the first terms of each binomial.<\/li>\n<li>Multiply the outer terms of the binomials.<\/li>\n<li>Multiply the inner terms of the binomials.<\/li>\n<li>Multiply the last terms of each binomial.<\/li>\n<li>Add the products.<\/li>\n<li>Combine like terms and simplify.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 5: Using FOIL to Multiply Binomials<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use FOIL to find the product.<\/p>\n<p>[latex](2x-18)(3x+3)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Find the product of the first terms.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1450\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5-300x32.webp\" alt=\"\" width=\"300\" height=\"32\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5-300x32.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5-65x7.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5-225x24.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5-350x37.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Find the product of the outer terms.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1451\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.1-300x32.webp\" alt=\"\" width=\"300\" height=\"32\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.1-300x32.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.1-65x7.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.1-225x24.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.1-350x37.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.1.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Find the product of the inner terms.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1452\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.2-300x32.webp\" alt=\"\" width=\"300\" height=\"32\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.2-300x32.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.2-65x7.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.2-225x24.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.2-350x37.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.2.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Find the product of the last terms.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1453\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.3-300x32.webp\" alt=\"\" width=\"300\" height=\"32\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.3-300x32.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.3-65x7.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.3-225x24.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.3-350x37.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.4.5.3.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>[latex]\\begin{array}{ll}6x^2+6x-54x-54 & \\quad \\text{Add the products.} \\\\6x^2+(6x-54x)-54 & \\quad \\text{Combine like terms.} \\\\6x^2-48x-54 & \\quad \\text{Simplify.} \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use FOIL to find the product.<\/p>\n<p>[latex](x+7)(3x-5)[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1167339428910\" data-depth=\"2\">\n<h3 data-type=\"title\">Perfect Square Trinomials<\/h3>\n<p id=\"fs-id1167339432924\">Certain binomial products have special forms. When a binomial is squared, the result is called a <span id=\"term-00019\" data-type=\"term\"><strong>perfect square trinomia<\/strong>l<\/span>. We can find the square by multiplying the binomial by itself. However, there is a special form that each of these perfect square trinomials takes, and memorizing the form makes squaring binomials much easier and faster. Let\u2019s look at a few perfect square trinomials to familiarize ourselves with the form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}\\, (x + 5)^2 &=& x^2 + 10x + 25 \\\\(x - 3)^2 &=& \\, x^2 - 6x + 9 \\\\(4x - 1)^2 &=& 16x^2 - 8x + 1\\end{array}[\/latex]<\/p>\n<p id=\"fs-id1167339224293\">Notice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of each trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms. Lastly, we see that the first sign of the trinomial is the same as the sign of the binomial.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Perfect Square Trinomials<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>When a binomial is squared, the result is the first term squared added to double the product of both terms and the last term squared.<\/p>\n<p style=\"text-align: center;\">[latex](x+a)^2=(x+a)(x+a)=x^2+2ax+a^2[\/latex]<\/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 id=\"fs-id1167339220533\"><strong>Given a binomial, square it using the formula for perfect square trinomials.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id1167339231805\" type=\"1\">\n<li>Square the first term of the binomial.<\/li>\n<li>Square the last term of the binomial.<\/li>\n<li>For the middle term of the trinomial, double the product of the two terms.<\/li>\n<li>Add and simplify.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 6: Expanding Perfect Squares<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Expand [latex](3x-8)^2.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Begin by squaring the first term and the last term. For the middle term of the trinomial, double the product of the two terms.<\/p>\n<p style=\"text-align: center;\">[latex](3x)^2-2(3x)(8)+(-8)^2[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]9x^2-48x+64[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #6<\/p>\n<\/header>\n<div class=\"textbox__content\">Expand [latex](4x-1)^2.[\/latex]<\/div>\n<\/div>\n<section id=\"fs-id1167339429838\" data-depth=\"2\">\n<h3 data-type=\"title\">Difference of Squares<\/h3>\n<p id=\"fs-id1167339188447\">Another special product is called the <strong><span id=\"term-00020\" data-type=\"term\">difference of squares<\/span>,<\/strong> which occurs when we multiply a binomial by another binomial with the same terms but the opposite sign. Let\u2019s see what happens when we multiply [latex](x+1)(x-1)[\/latex] using the FOIL method.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}\\, (x+1)(x-1) &=& x^2-x+x-1 \\\\ &=& \\, x^2-1 \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1167339432898\">The middle term drops out, resulting in a difference of squares. Just as we did with the perfect squares, let\u2019s look at a few examples.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}\\, (x+5)(x-5) &=& x^2-25 \\\\ (x+11)(x-11) &=& \\, x^2-121 \\\\ (2x+3)(2x-3) &=& 4x^2-9 \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1167339220812\">Because the sign changes in the second binomial, the outer and inner terms cancel each other out, and we are left only with the square of the first term minus the square of the last term.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1167339220824\"><strong>Q: Is there a special form for the sum of squares?<\/strong><\/p>\n<p id=\"fs-id1167339199657\"><em data-effect=\"italics\">A: No. The difference of squares occurs because the opposite signs of the binomials cause the middle terms to disappear. There are no two binomials that multiply to equal a sum of squares.<\/em><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Difference of Squares<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>When a binomial is multiplied by a binomial with the same terms separated by the opposite sign, the result is the square of the first term minus the square of the last term.<\/p>\n<p style=\"text-align: center;\">[latex](a+b)(a-b)=a^2-b^2[\/latex]<\/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 id=\"fs-id1167339273978\"><strong>Given a binomial multiplied by a binomial with the same terms but the opposite sign, find the difference of squares.<\/strong><\/p>\n<ol id=\"fs-id1167339273983\" type=\"1\">\n<li>Square the first term of the binomials.<\/li>\n<li>Square the last term of the binomials.<\/li>\n<li>Subtract the square of the last term from the square of the first term.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 7: Multiplying Binomials Resulting in a Difference of Squares<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Multiply [latex](9x+4)(9x-4).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Square the first term to get [latex](9x)^2=81x^2.[\/latex] Square the last term to get [latex]4^2=16.[\/latex] Subtract the square of the last term from the square of the first term to find the product of [latex]81x^2-16.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Multiply [latex](2x+7)(2x-7).[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1167339138125\" data-depth=\"1\">\n<h2 data-type=\"title\">Performing Operations with Polynomials of Several Variables<\/h2>\n<p id=\"fs-id1167339138130\">We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}(a + 2b)(4a - b - c) & \\\\a(4a - b - c) + 2b(4a - b - c) & \\quad \\text{Use the distributive property.} \\\\4a^2 - ab - ac + 8ab - 2b^2 - 2bc & \\quad \\text{Multiply.} \\\\4a^2 + (-ab + 8ab) - ac - 2b^2 - 2bc & \\quad \\text{Combine like terms.} \\\\4a^2 + 7ab - ac - 2bc - 2b^2 & \\quad \\text{Simplify.}\\end{array}[\/latex]<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 8: Multiplying Polynomials Containing Several Variables<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Multiply [latex](x+4)(3x-2y+5).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Follow the same steps that we used to multiply polynomials containing only one variable.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}x(3x-2y+5)+4_3x-2y+5) & \\quad \\text{Use the distributive property.} \\\\3x^2-2xy+5x+12x-8y+20 & \\quad \\text{Multiply.} \\\\3x^2-2xy+(5x+12x)-8y+20 & \\quad \\text{Combine like terms.} \\\\3x^2-2xy+17x-8y+20 & \\quad \\text{Simplify.}\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Multiply [latex](3x-1)(2x+7y-9).[\/latex]<\/p>\n<\/div>\n<\/div>\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 polynomials.<\/p>\n<ul>\n<li><a href=\"http:\/\/openstax.org\/l\/addsubpoly\" target=\"_blank\" rel=\"noopener nofollow\">Adding and Subtracting Polynomials<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/multiplpoly\" target=\"_blank\" rel=\"noopener nofollow\">Multiplying Polynomials<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/specialpolyprod\" target=\"_blank\" rel=\"noopener nofollow\">Special Products of Polynomials<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">1.4 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1167339184236\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1167339184243\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1167339184249\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339184250\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339184249-solution\">1<\/a><span class=\"os-divider\">. <\/span>Evaluate the following statement: The degree of a polynomial in standard form is the exponent of the leading term. Explain why the statement is true or false.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339216138\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339216140\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>Many times, multiplying two binomials with two variables results in a trinomial. This is not the case when there is a difference of two squares. Explain why the product in this case is also a binomial.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339216146\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339216147\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339216146-solution\">3<\/a><span class=\"os-divider\">. <\/span>You can multiply polynomials with any number of terms and any number of variables using four basic steps over and over until you reach the expanded polynomial. What are the four steps?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339216157\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339216158\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>State whether the following statement is true and explain why or why not: A trinomial is always a higher degree than a monomial.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1167339216165\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id1167339216170\">For the following exercises, identify the degree of the polynomial.<\/p>\n<div id=\"fs-id1167339216173\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339216174\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339216173-solution\">5<\/a><span class=\"os-divider\">. <\/span> [latex]7x-2x^2+13[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339299774\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339299775\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex]14m^3+m^2-16m+8[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339299820\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339299821\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339299820-solution\">7<\/a><span class=\"os-divider\">.<\/span> [latex]-625a^8+16b^4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339268846\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339268847\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">.<\/span> [latex]200p-30p^2m+40m^3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339223145\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339223146\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339223145-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex]x^2+4x+4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339223181\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339223182\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex]6y^4-y^5+3y-4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1167339432583\">For the following exercises, find the sum or difference.<\/p>\n<div id=\"fs-id1167339432587\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339432588\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339432587-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex](12x^2+3x)-(8x^2-19)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339185745\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339185746\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex](4z^3+8z^2-z)+(-2z^2+z+6)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339306370\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339306371\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339306370-solution\">13<\/a><span class=\"os-divider\">.<\/span> [latex](6w^2+24w+24)-(3w^2-6w+3)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339223833\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339223834\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex](7a^3+6a^2-4a-13)+(_3a^3-4a^2+6a+17)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339262640\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339262641\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339262640-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex](11b^4-6b^3+18b^2-4b+8)-(3b^3+6b^2+3b)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339196660\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339196661\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex]- (49p^2-25)+(16p^4-32p^2+16)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1167339196741\">For the following exercises, find the product.<\/p>\n<div id=\"fs-id1167339196745\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339196746\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339196745-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex](4x+2)(6x-4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339259698\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339259699\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex](14c^2+4c)(2c^2-3c)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339223391\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339223392\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339223391-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex](6b^2-6)(4b^2-4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339223497\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339223498\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex](3d-5)(2d+9)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339306553\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339306554\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339306553-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex](9v-11)(11v-9)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339286443\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339286444\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex](4t^2+7t)(-3t^2+4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339286513\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339286514\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339286513-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex](8n-4)(n^2+9)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1167339199464\">For the following exercises, expand the binomial.<\/p>\n<div id=\"fs-id1167339199468\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339199469\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">.<\/span> [latex](4x+5)^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339199508\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339199509\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339199508-solution\">25<\/a><span class=\"os-divider\">.<\/span> [latex](3y-7)^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339220230\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339220232\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">.<\/span> [latex](12-4x)^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339220271\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339220272\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339220271-solution\">27<\/a><span class=\"os-divider\">.<\/span> [latex](4p+9)^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339220345\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339220346\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">.<\/span> [latex](2m-3)^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339226325\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339226326\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339226325-solution\">29<\/a><span class=\"os-divider\">.<\/span> [latex](3y-6)^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339226399\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339226400\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">.<\/span> [latex](9b+1)^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1167339196445\">For the following exercises, multiply the binomials.<\/p>\n<div id=\"fs-id1167339196448\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339196449\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339196448-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex](4c+1)(4c-1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339196525\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339196526\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex](9a-4)(9a+4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339196575\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339196576\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339196575-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex](15n-6)(15n+6)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339260474\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339260475\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex](25b+20(25b-2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339260524\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339260525\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339260524-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex](4+4m)(4-4m)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339239069\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339239070\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex](14p+7)(14p-7)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339239119\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339239120\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339239119-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex](11q-10)(11q+10)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1167339321358\">For the following exercises, multiply the polynomials.<\/p>\n<div id=\"fs-id1167339321361\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339321362\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex](2x^2+2x+1)(4x-1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339321425\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339321426\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339321425-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex](4t^2+t-7)(4t^2-1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339138836\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339138837\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex](x-1)(x^2-2x+1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339138896\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339138897\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339138896-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex](y-2)(y^2-4y-9)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339273830\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339273831\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex](6k-5)(6k^2+5k-1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339273894\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339273895\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339273894-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex](3p^2+2p-10)(p-1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339344452\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339344453\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex](4m-13)(2m^2-7m+9)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339344516\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339344517\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339344516-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex](a+b)(a-b)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339213948\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339213949\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex](4x-6y)(6x-4y)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339214002\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339214003\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339214002-solution\">47<\/a><span class=\"os-divider\">.<\/span> [latex](4t-5u)^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339214089\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339214090\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex](9m+4n-1)(2m+8)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339222871\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339222872\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339222871-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex](4t-x)(t-x+1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339222977\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339222978\" data-type=\"problem\">\n<p><span class=\"os-number\">50<\/span><span class=\"os-divider\">.<\/span> [latex](b^2-1)(a^2+2ab+b^2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339223053\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339223054\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339223053-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex](4r-d)(6r+7d)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339225658\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339225659\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">.<\/span> [latex](x+y)(x^2-xy+y^2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1167339225725\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<div id=\"fs-id1167339225730\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339225731\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339225730-solution\">53<\/a><span class=\"os-divider\">. <\/span>A developer wants to purchase a plot of land to build a house. The area of the plot can be described by the following expression: [latex](4x+1)(8x-3)[\/latex] where <em data-effect=\"italics\">x<\/em> is measured in meters. Multiply the binomials to find the area of the plot in standard form.<\/p>\n<div class=\"os-problem-container\"><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339242981\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339242982\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>A prospective buyer wants to know how much grain a specific silo can hold. The area of the floor of the silo is [latex](2x+9)^2.[\/latex] The height of the silo is [latex]10x+10,[\/latex] where <em data-effect=\"italics\">x<\/em> is measured in feet. Expand the square and multiply by the height to find the expression that shows how much grain the silo can hold.<\/p>\n<div class=\"os-problem-container\"><\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1167339243059\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<p id=\"fs-id1167339243064\">For the following exercises, perform the given operations.<\/p>\n<div id=\"fs-id1167339243067\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339243068\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339243067-solution\">55<\/a><span class=\"os-divider\">. <\/span> [latex](4t-7)^2(2t+1)-(4t^2+2t+11)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339230814\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339230815\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span> [latex](3b+6)(3b-6)(9b^2-36)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339230891\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1167339230892\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339230891-solution\">57<\/a><span class=\"os-divider\">. <\/span> [latex](a^2+4ac+4c^2)(a^2-4c^2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n","protected":false},"author":158,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-87","chapter","type-chapter","status-publish","hentry"],"part":27,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/87","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":11,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/87\/revisions"}],"predecessor-version":[{"id":1801,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/87\/revisions\/1801"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/27"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/87\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=87"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=87"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=87"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=87"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}