{"id":1587,"date":"2022-04-15T00:42:20","date_gmt":"2022-04-15T00:42:20","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=1587"},"modified":"2026-01-06T18:09:10","modified_gmt":"2026-01-06T18:09:10","slug":"3-5-1-factoring-polynomials-the-gcf-and-grouping-methods","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/3-5-1-factoring-polynomials-the-gcf-and-grouping-methods\/","title":{"raw":"3.5.1: Factoring Polynomials - GCF and Grouping","rendered":"3.5.1: Factoring Polynomials &#8211; GCF and Grouping"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" style=\"text-align: center;\" role=\"main\">\r\n<div id=\"post-146\" class=\"standard post-146 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Describe the meaning of greatest common factor<\/li>\r\n \t<li>Determine the greatest common factor of multiple terms in a polynomial<\/li>\r\n \t<li>Factor a polynomial function using GCF method<\/li>\r\n \t<li>Factor a polynomial by grouping<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Factors and the Greatest Common Factor<\/h2>\r\nA<strong><em> factor\u00a0<\/em><\/strong>of a number is any number that divides exactly into the number. For example, only the numbers 1, 2, 4, 5, 10, and 20 all divide exactly into 20 with no remainder. Consequently, the factors of 20 are\u00a01, 2, 4, 5, 10, and 20. A\u00a0<em><strong>factor\u00a0<\/strong><\/em>of an algebraic term is any number, variable, or the combination of number and variable that divides exactly into the term. For example, [latex]6x^2[\/latex] is divisible by [latex]1,\\,2,\\,3,\\,6,\\,x,\\,2x,\\,3x,\\,6x,\\,x^2,\\,2x^2,\\,3x^2,[\/latex] and [latex]6x^2[\/latex]. So each of these are factors of [latex]6x^2[\/latex].\r\n\r\nTo factor a term is to rewrite it as a product of factors. For example, [latex]20=4\\cdot{5}[\/latex] or [latex]6x^2=2x\\cdot 3x[\/latex]. In algebra, we use the word factor as both a noun ([latex]2x[\/latex] is a factor of [latex]6x^2[\/latex]) and as a verb (factor [latex]6x^2[\/latex], i.e. [latex]6x^2=2\\cdot 3\\cdot x\\cdot x[\/latex]).\u00a0<strong><em>Factoring<\/em>\u00a0<\/strong>can also turn addition or subtraction into a product and\u00a0is very helpful in simplifying polynomial functions and solving equations involving\u00a0polynomial functions.\r\n\r\nThe\u00a0<strong><em>greatest common factor<\/em>\u00a0<\/strong>(GCF) of two numbers is the largest number that divides exactly into both numbers. For instance, [latex]4[\/latex] is the GCF of [latex]16[\/latex] and [latex]20[\/latex] because it is the largest number that divides exactly into both [latex]16[\/latex] and [latex]20[\/latex]. The GCF of terms works the same way: [latex]4x[\/latex] is the GCF of [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex] because it is the largest polynomial that divides exactly into both [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex].\r\n\r\nWhen factoring a polynomial function, our first step should always be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables. The GCF of the variables will always have the smallest exponent of each variable.\r\n<div class=\"textbox\">\r\n<h3>Greatest Common Factor<\/h3>\r\nThe\u00a0<strong>greatest common factor\u00a0<\/strong>(GCF) of a group of terms is the largest term that divides exactly into the polynomials.\r\n\r\n<\/div>\r\nWe can use prime factorization to find the GCF.\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nFind the greatest common factor of [latex]25b^{3}[\/latex] and [latex]10b^{2}[\/latex].\r\n<h4>Solution<\/h4>\r\n[latex]\\begin{array}{l}\\,\\,25b^{3}=\\color{blue}{5}\\cdot5\\cdot{\\color{blue}{b}}\\cdot{\\color{blue}{b}}\\cdot{b}\\\\\\,\\,10b^{2}=2\\cdot\\color{blue}{5}\\cdot{\\color{blue}{b}}\\cdot{\\color{blue}{b}}\\\\\\text{GCF}=\\color{blue}{5}\\cdot{\\color{blue}{b}}\\cdot{\\color{blue}{b}}\\;\\;\\;\\;\\text{Choose factors that appear in each prime factorization.}\\\\\\text{GCF}=5b^{2}\\end{array}[\/latex]\r\n\r\n&nbsp;\r\n\r\n[latex]25b^3[\/latex] and [latex]10b^2[\/latex]have the factors\u00a0[latex]5,\\;b[\/latex], and [latex]b[\/latex] in common, which means their greatest common factor is [latex]5\\cdot{b}\\cdot{b}=5b^{2}[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nSometimes we encounter a polynomial with more than one variable, so it is important to check whether both variables are part of the GCF. In the next example, we find the GCF of two terms which each contain two variables.\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nFind the greatest common factor of [latex]81c^{3}d[\/latex] and [latex]45c^{2}d^{2}[\/latex].\r\n<h4>Solution<\/h4>\r\nLook for factors that are common to each term:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,81c^3d=\\color{blue}{3}\\cdot\\color{blue}{3}\\cdot3\\cdot3\\cdot{\\color{blue}{c}}\\cdot{\\color{blue}{c}}\\cdot{c}\\cdot{\\color{blue}{d}}\\\\45c^{2}d^{2}=\\color{blue}{3}\\cdot\\color{blue}{3}\\cdot5\\cdot{\\color{blue}{c}}\\cdot{\\color{blue}{c}}\\cdot{\\color{blue}{d}}\\cdot{d}\\\\\\,\\,\\,\\,\\text{GCF}=\\color{blue}{3}\\cdot\\color{blue}{3}\\cdot{\\color{blue}{c}}\\cdot{\\color{blue}{c}}\\cdot{\\color{blue}{d}}\\\\\\,\\,\\,\\,\\text{GCF}=9c^{2}d\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nDetermine the greatest common factor of the terms:\r\n<ol>\r\n \t<li>\u00a0[latex]14x^2y^3[\/latex];\u00a0 [latex]21x^3y^2[\/latex]<\/li>\r\n \t<li>\u00a0[latex]18x^2y[\/latex]; [latex]27xy[\/latex]<\/li>\r\n \t<li>\u00a0[latex]24x^2y^5[\/latex]; [latex]36x^2y^3[\/latex]; [latex]60x^3y^4[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm874\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm874\"]\r\n<ol>\r\n \t<li>[latex]7x^2y^2[\/latex]<\/li>\r\n \t<li>[latex]9xy[\/latex]<\/li>\r\n \t<li>[latex]12x^2y^3[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Write a Function in Factored Form using the GCF<\/h2>\r\nNow that we have practiced identifying the GCF of terms with one and two variables, we can apply this idea to factoring\u00a0the GCF out of a polynomial function. Notice that\u00a0the instructions are now \u201cFactor\u201d instead of \u201cFind the greatest common factor.\u201d\r\n\r\nTo write a polynomial function in factored form, first identify the greatest common factor of the terms. We can then use the distributive property to rewrite the polynomial in factored form. Recall that the\u00a0<strong><em>distributive property of multiplication over addition<\/em>\u00a0<\/strong>states that [latex]a(b+c)=ab+ac[\/latex] for all terms [latex]a,\\;b,[\/latex] and [latex]c[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nWrite the following function in factored form: [latex]p(b)=25b^{3}+10b^{2}[\/latex].\r\n<h4>Solution<\/h4>\r\nThe GCF of [latex]25b^{3}[\/latex] and [latex]10b^{2}[\/latex] is [latex]5b^{2}[\/latex].\r\n<p style=\"text-align: left;\">Use the distributive property to \"pull out\" the GCF from all terms in the function by first rewriting each term with the GCF as a factor:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}p(b)&amp;=25b^3+10b^2\\\\ &amp;=\\color{blue}{5b^2}\\cdot 5b+\\color{blue}{5b^2}\\cdot 2\\\\ &amp;=\\color{blue}{5b^2} \\left (5b+2\\right )\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\nThe factored form of the polynomial [latex]25b^{3}+10b^{2}[\/latex] is [latex]5b^{2}\\left(5b+2\\right)[\/latex]. We can check this by distributing [latex]5b^2[\/latex]:\u00a0 [latex]5b^{2}\\left(5b+2\\right)=25b^{3}+10b^{2}[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nWrite the following function in factored form: [latex]f(x)=12x^3-16x^2+20x[\/latex].\r\n<h4>Solution<\/h4>\r\nThe GCF of\u00a0[latex]12x^3,\\;16x^2\\;,20x=4x[\/latex].\r\n\r\nWrite each term in the function with the GCF as a factor, then \"pull out\" the GCF using the distributive property.\r\n\r\n[latex]\\begin{aligned}f(x)&amp;=12x^3-16x^2+20x\\\\&amp;=\\color{blue}{4x}\\cdot 3x^2-\\color{blue}{4x}\\cdot 4x+\\color{blue}{4x}\\cdot 5\\\\&amp;=\\color{blue}{4x}\\left (3x^2-4x+5\\right )\\end{aligned}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nWrite the following functions in factored form:\r\n<ol>\r\n \t<li>\u00a0[latex]f(x)=9x^4-36x^3+45x^2[\/latex]<\/li>\r\n \t<li>\u00a0[latex]g(x)=15x^5-30x^3-45x^2[\/latex]<\/li>\r\n \t<li>\u00a0[latex]p(x)=18x^7-36x^5+54x^3[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm564\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm564\"]\r\n<ol>\r\n \t<li>[latex]f(x)=9x^2\\left (x^2-4x+5\\right )[\/latex]<\/li>\r\n \t<li>[latex]g(x)=15x^2\\left (x^3-2x-3\\right )[\/latex]<\/li>\r\n \t<li>[latex]p(x)=18x^3\\left (x^4-2x^2+3\\right )[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p role=\"contentinfo\">When the leading coefficient of a polynomial function is negative, we include the negative sign with the GCF.<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 5<\/h3>\r\nWrite the following function in factored form: [latex]p(x)=-14x^3+21x^2[\/latex].\r\n<h4>Solution<\/h4>\r\nThe GCF = [latex]7x^2[\/latex] but we \"pull out\" [latex]-7x^2[\/latex] since the leading coefficient is negative.\r\n\r\nWrite each term in [latex]p(x)[\/latex] using\u00a0[latex]-7x^2[\/latex] as a factor:\r\n\r\n[latex]\\begin{aligned}-14x^3&amp;=-7x^2\\cdot 2x\\\\ \\\\21x^2&amp;=-7x^2\\cdot (-3)\\end{aligned}[\/latex]\r\n\r\n&nbsp;\r\n\r\nSo, [latex]p(x)=-7x^2\\left (2x-3\\right )[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nWrite the following function in factored form: [latex]p(x)=-8x^4+6x^3-10x^2[\/latex]\r\n\r\n[reveal-answer q=\"hjm969\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm969\"]\r\n\r\n[latex]p(x)=-2x^2\\left (4x^2-3x+5\\right )[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 class=\"citations-section\" role=\"contentinfo\">Write a Polynomial Function in Factored Form using Grouping<\/h2>\r\nWhen it appears that there is no common factor, except 1, among the terms of a polynomial function, there may be a common factor that is embedded in the polynomial and cannot be seen from the surface.\u00a0<span style=\"font-size: medium;\">For example, the polynomial function [latex]f(x)=5x^3-20x^2+3x-12[\/latex] has no common factor (except 1) across its four terms. However, if we group the first two terms and the last two terms of the polynomial, each pair has a common factor. There is a common factor [latex]5x^2[\/latex] in the first group and a common factor [latex]3[\/latex] in the second group.\u00a0<\/span>\r\n<p style=\"padding-left: 240px;\">[latex]\\begin{aligned}f(x)&amp;=\\color{blue}{5x^3-20x^2}+\\color{green}{3x-12}\\\\ \\\\&amp;=\\color{blue}{5x^2}\\color{red}{(x-4)}+\\color{green}{3}\\color{red}{(x-4)}\\\\ \\\\&amp;=\\color{red}{(x-4)}\\left (\\color{blue}{5x^2}+\\color{green}{3}\\right )\\end{aligned}[\/latex]<\/p>\r\nIf we factor out these common factors from each group, It turns out that we find a common factor [latex]\\color{red}{(x-4)}[\/latex] across the two groups. After we factor out this common factor, we obtain the factored form [latex]f(x)=(x-4)(5x^2+3)[\/latex].\r\n\r\nThis method of factoring is called <em style=\"font-size: 1rem; orphans: 1; text-align: initial;\"><strong>factoring by<\/strong> <strong>grouping<\/strong><\/em><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">.\u00a0It\u00a0does not work for every polynomial, but it can be very useful when it does.<\/span>\r\n<div class=\"textbox examples\">\r\n<h3>Example 6<\/h3>\r\nWrite the following function in factored form: [latex]g(x)=5a^2x+2a^2y-5bx-2by[\/latex].\r\n<h4>Solution<\/h4>\r\nThere is no common factor (other than 1) for all four terms of the function, so we will try factoring by grouping.\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}g(x)&amp;=5\\color{blue}{a^2}x+2\\color{blue}{a^2}y\\color{green}{-}5\\color{green}{b}x\\color{green}{-}2\\color{green}{b}y\\\\ \\\\&amp;=\\color{blue}{a^2}\\color{red}{(5x+2y)}\\color{green}{-b}\\color{red}{(5x+2y)}\\\\ \\\\&amp;=\\color{red}{(5x+2y)}(\\color{blue}{a^2}\\color{green}{-b})\\end{aligned}[\/latex]<\/p>\r\nNotice that since the third term is negative, we factor out the negative sign along with [latex]b[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nWrite the following function in factored form: [latex]f(x)=4ax^2+7a^2cx-4bx-7abc[\/latex].\r\n\r\n[reveal-answer q=\"hjm792\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm792\"]\r\n\r\n[latex](4x+7ac)(ax-b)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIt is not necessary to always group the first two terms and the last two terms. In the next example, we will group the first and third terms, and the second and fourth terms. The next example has the same function as the last example, but we will group the terms differently.\r\n<div class=\"textbox examples\">\r\n<h3>Example 7<\/h3>\r\nWrite the following function in factored form: [latex]g(x)=5a^2x+2a^2y-5bx-2by[\/latex].\r\n<h4>Solution<\/h4>\r\n[latex]\\begin{aligned}g(x)&amp;=5a^2x+2a^2y-5bx-2by\\\\ \\\\&amp;=5a^2x-5bx+2a^2y-2by\\\\ \\\\&amp;=5x(a^2-b)+2y(a^2-b)\\\\ \\\\&amp;=(a^2-b)(5x+2y)\\end{aligned}[\/latex]\r\n\r\nNotice that this is the same answer as the previous example with the factors in reverse order.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nWrite the following function in factored form: [latex]f(x)=4ax^2+7a^2cx-4bx-7abc[\/latex] by grouping the 1st and 3rd terms, and the 2nd and 4th terms.\r\n\r\n[reveal-answer q=\"hjm863\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm863\"]\r\n\r\n[latex](ax-b)(4x+7ac)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" style=\"text-align: center;\" role=\"main\">\n<div id=\"post-146\" class=\"standard post-146 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Describe the meaning of greatest common factor<\/li>\n<li>Determine the greatest common factor of multiple terms in a polynomial<\/li>\n<li>Factor a polynomial function using GCF method<\/li>\n<li>Factor a polynomial by grouping<\/li>\n<\/ul>\n<\/div>\n<h2>Factors and the Greatest Common Factor<\/h2>\n<p>A<strong><em> factor\u00a0<\/em><\/strong>of a number is any number that divides exactly into the number. For example, only the numbers 1, 2, 4, 5, 10, and 20 all divide exactly into 20 with no remainder. Consequently, the factors of 20 are\u00a01, 2, 4, 5, 10, and 20. A\u00a0<em><strong>factor\u00a0<\/strong><\/em>of an algebraic term is any number, variable, or the combination of number and variable that divides exactly into the term. For example, [latex]6x^2[\/latex] is divisible by [latex]1,\\,2,\\,3,\\,6,\\,x,\\,2x,\\,3x,\\,6x,\\,x^2,\\,2x^2,\\,3x^2,[\/latex] and [latex]6x^2[\/latex]. So each of these are factors of [latex]6x^2[\/latex].<\/p>\n<p>To factor a term is to rewrite it as a product of factors. For example, [latex]20=4\\cdot{5}[\/latex] or [latex]6x^2=2x\\cdot 3x[\/latex]. In algebra, we use the word factor as both a noun ([latex]2x[\/latex] is a factor of [latex]6x^2[\/latex]) and as a verb (factor [latex]6x^2[\/latex], i.e. [latex]6x^2=2\\cdot 3\\cdot x\\cdot x[\/latex]).\u00a0<strong><em>Factoring<\/em>\u00a0<\/strong>can also turn addition or subtraction into a product and\u00a0is very helpful in simplifying polynomial functions and solving equations involving\u00a0polynomial functions.<\/p>\n<p>The\u00a0<strong><em>greatest common factor<\/em>\u00a0<\/strong>(GCF) of two numbers is the largest number that divides exactly into both numbers. For instance, [latex]4[\/latex] is the GCF of [latex]16[\/latex] and [latex]20[\/latex] because it is the largest number that divides exactly into both [latex]16[\/latex] and [latex]20[\/latex]. The GCF of terms works the same way: [latex]4x[\/latex] is the GCF of [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex] because it is the largest polynomial that divides exactly into both [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex].<\/p>\n<p>When factoring a polynomial function, our first step should always be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables. The GCF of the variables will always have the smallest exponent of each variable.<\/p>\n<div class=\"textbox\">\n<h3>Greatest Common Factor<\/h3>\n<p>The\u00a0<strong>greatest common factor\u00a0<\/strong>(GCF) of a group of terms is the largest term that divides exactly into the polynomials.<\/p>\n<\/div>\n<p>We can use prime factorization to find the GCF.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Find the greatest common factor of [latex]25b^{3}[\/latex] and [latex]10b^{2}[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>[latex]\\begin{array}{l}\\,\\,25b^{3}=\\color{blue}{5}\\cdot5\\cdot{\\color{blue}{b}}\\cdot{\\color{blue}{b}}\\cdot{b}\\\\\\,\\,10b^{2}=2\\cdot\\color{blue}{5}\\cdot{\\color{blue}{b}}\\cdot{\\color{blue}{b}}\\\\\\text{GCF}=\\color{blue}{5}\\cdot{\\color{blue}{b}}\\cdot{\\color{blue}{b}}\\;\\;\\;\\;\\text{Choose factors that appear in each prime factorization.}\\\\\\text{GCF}=5b^{2}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>[latex]25b^3[\/latex] and [latex]10b^2[\/latex]have the factors\u00a0[latex]5,\\;b[\/latex], and [latex]b[\/latex] in common, which means their greatest common factor is [latex]5\\cdot{b}\\cdot{b}=5b^{2}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Sometimes we encounter a polynomial with more than one variable, so it is important to check whether both variables are part of the GCF. In the next example, we find the GCF of two terms which each contain two variables.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Find the greatest common factor of [latex]81c^{3}d[\/latex] and [latex]45c^{2}d^{2}[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>Look for factors that are common to each term:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,81c^3d=\\color{blue}{3}\\cdot\\color{blue}{3}\\cdot3\\cdot3\\cdot{\\color{blue}{c}}\\cdot{\\color{blue}{c}}\\cdot{c}\\cdot{\\color{blue}{d}}\\\\45c^{2}d^{2}=\\color{blue}{3}\\cdot\\color{blue}{3}\\cdot5\\cdot{\\color{blue}{c}}\\cdot{\\color{blue}{c}}\\cdot{\\color{blue}{d}}\\cdot{d}\\\\\\,\\,\\,\\,\\text{GCF}=\\color{blue}{3}\\cdot\\color{blue}{3}\\cdot{\\color{blue}{c}}\\cdot{\\color{blue}{c}}\\cdot{\\color{blue}{d}}\\\\\\,\\,\\,\\,\\text{GCF}=9c^{2}d\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Determine the greatest common factor of the terms:<\/p>\n<ol>\n<li>\u00a0[latex]14x^2y^3[\/latex];\u00a0 [latex]21x^3y^2[\/latex]<\/li>\n<li>\u00a0[latex]18x^2y[\/latex]; [latex]27xy[\/latex]<\/li>\n<li>\u00a0[latex]24x^2y^5[\/latex]; [latex]36x^2y^3[\/latex]; [latex]60x^3y^4[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm874\">Show Answer<\/span><\/p>\n<div id=\"qhjm874\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]7x^2y^2[\/latex]<\/li>\n<li>[latex]9xy[\/latex]<\/li>\n<li>[latex]12x^2y^3[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Write a Function in Factored Form using the GCF<\/h2>\n<p>Now that we have practiced identifying the GCF of terms with one and two variables, we can apply this idea to factoring\u00a0the GCF out of a polynomial function. Notice that\u00a0the instructions are now \u201cFactor\u201d instead of \u201cFind the greatest common factor.\u201d<\/p>\n<p>To write a polynomial function in factored form, first identify the greatest common factor of the terms. We can then use the distributive property to rewrite the polynomial in factored form. Recall that the\u00a0<strong><em>distributive property of multiplication over addition<\/em>\u00a0<\/strong>states that [latex]a(b+c)=ab+ac[\/latex] for all terms [latex]a,\\;b,[\/latex] and [latex]c[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Write the following function in factored form: [latex]p(b)=25b^{3}+10b^{2}[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>The GCF of [latex]25b^{3}[\/latex] and [latex]10b^{2}[\/latex] is [latex]5b^{2}[\/latex].<\/p>\n<p style=\"text-align: left;\">Use the distributive property to &#8220;pull out&#8221; the GCF from all terms in the function by first rewriting each term with the GCF as a factor:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}p(b)&=25b^3+10b^2\\\\ &=\\color{blue}{5b^2}\\cdot 5b+\\color{blue}{5b^2}\\cdot 2\\\\ &=\\color{blue}{5b^2} \\left (5b+2\\right )\\end{aligned}[\/latex]<\/p>\n<\/div>\n<p>The factored form of the polynomial [latex]25b^{3}+10b^{2}[\/latex] is [latex]5b^{2}\\left(5b+2\\right)[\/latex]. We can check this by distributing [latex]5b^2[\/latex]:\u00a0 [latex]5b^{2}\\left(5b+2\\right)=25b^{3}+10b^{2}[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>Write the following function in factored form: [latex]f(x)=12x^3-16x^2+20x[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>The GCF of\u00a0[latex]12x^3,\\;16x^2\\;,20x=4x[\/latex].<\/p>\n<p>Write each term in the function with the GCF as a factor, then &#8220;pull out&#8221; the GCF using the distributive property.<\/p>\n<p>[latex]\\begin{aligned}f(x)&=12x^3-16x^2+20x\\\\&=\\color{blue}{4x}\\cdot 3x^2-\\color{blue}{4x}\\cdot 4x+\\color{blue}{4x}\\cdot 5\\\\&=\\color{blue}{4x}\\left (3x^2-4x+5\\right )\\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Write the following functions in factored form:<\/p>\n<ol>\n<li>\u00a0[latex]f(x)=9x^4-36x^3+45x^2[\/latex]<\/li>\n<li>\u00a0[latex]g(x)=15x^5-30x^3-45x^2[\/latex]<\/li>\n<li>\u00a0[latex]p(x)=18x^7-36x^5+54x^3[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm564\">Show Answer<\/span><\/p>\n<div id=\"qhjm564\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]f(x)=9x^2\\left (x^2-4x+5\\right )[\/latex]<\/li>\n<li>[latex]g(x)=15x^2\\left (x^3-2x-3\\right )[\/latex]<\/li>\n<li>[latex]p(x)=18x^3\\left (x^4-2x^2+3\\right )[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p role=\"contentinfo\">When the leading coefficient of a polynomial function is negative, we include the negative sign with the GCF.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 5<\/h3>\n<p>Write the following function in factored form: [latex]p(x)=-14x^3+21x^2[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>The GCF = [latex]7x^2[\/latex] but we &#8220;pull out&#8221; [latex]-7x^2[\/latex] since the leading coefficient is negative.<\/p>\n<p>Write each term in [latex]p(x)[\/latex] using\u00a0[latex]-7x^2[\/latex] as a factor:<\/p>\n<p>[latex]\\begin{aligned}-14x^3&=-7x^2\\cdot 2x\\\\ \\\\21x^2&=-7x^2\\cdot (-3)\\end{aligned}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>So, [latex]p(x)=-7x^2\\left (2x-3\\right )[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Write the following function in factored form: [latex]p(x)=-8x^4+6x^3-10x^2[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm969\">Show Answer<\/span><\/p>\n<div id=\"qhjm969\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]p(x)=-2x^2\\left (4x^2-3x+5\\right )[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2 class=\"citations-section\" role=\"contentinfo\">Write a Polynomial Function in Factored Form using Grouping<\/h2>\n<p>When it appears that there is no common factor, except 1, among the terms of a polynomial function, there may be a common factor that is embedded in the polynomial and cannot be seen from the surface.\u00a0<span style=\"font-size: medium;\">For example, the polynomial function [latex]f(x)=5x^3-20x^2+3x-12[\/latex] has no common factor (except 1) across its four terms. However, if we group the first two terms and the last two terms of the polynomial, each pair has a common factor. There is a common factor [latex]5x^2[\/latex] in the first group and a common factor [latex]3[\/latex] in the second group.\u00a0<\/span><\/p>\n<p style=\"padding-left: 240px;\">[latex]\\begin{aligned}f(x)&=\\color{blue}{5x^3-20x^2}+\\color{green}{3x-12}\\\\ \\\\&=\\color{blue}{5x^2}\\color{red}{(x-4)}+\\color{green}{3}\\color{red}{(x-4)}\\\\ \\\\&=\\color{red}{(x-4)}\\left (\\color{blue}{5x^2}+\\color{green}{3}\\right )\\end{aligned}[\/latex]<\/p>\n<p>If we factor out these common factors from each group, It turns out that we find a common factor [latex]\\color{red}{(x-4)}[\/latex] across the two groups. After we factor out this common factor, we obtain the factored form [latex]f(x)=(x-4)(5x^2+3)[\/latex].<\/p>\n<p>This method of factoring is called <em style=\"font-size: 1rem; orphans: 1; text-align: initial;\"><strong>factoring by<\/strong> <strong>grouping<\/strong><\/em><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">.\u00a0It\u00a0does not work for every polynomial, but it can be very useful when it does.<\/span><\/p>\n<div class=\"textbox examples\">\n<h3>Example 6<\/h3>\n<p>Write the following function in factored form: [latex]g(x)=5a^2x+2a^2y-5bx-2by[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>There is no common factor (other than 1) for all four terms of the function, so we will try factoring by grouping.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}g(x)&=5\\color{blue}{a^2}x+2\\color{blue}{a^2}y\\color{green}{-}5\\color{green}{b}x\\color{green}{-}2\\color{green}{b}y\\\\ \\\\&=\\color{blue}{a^2}\\color{red}{(5x+2y)}\\color{green}{-b}\\color{red}{(5x+2y)}\\\\ \\\\&=\\color{red}{(5x+2y)}(\\color{blue}{a^2}\\color{green}{-b})\\end{aligned}[\/latex]<\/p>\n<p>Notice that since the third term is negative, we factor out the negative sign along with [latex]b[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>Write the following function in factored form: [latex]f(x)=4ax^2+7a^2cx-4bx-7abc[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm792\">Show Answer<\/span><\/p>\n<div id=\"qhjm792\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex](4x+7ac)(ax-b)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>It is not necessary to always group the first two terms and the last two terms. In the next example, we will group the first and third terms, and the second and fourth terms. The next example has the same function as the last example, but we will group the terms differently.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 7<\/h3>\n<p>Write the following function in factored form: [latex]g(x)=5a^2x+2a^2y-5bx-2by[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>[latex]\\begin{aligned}g(x)&=5a^2x+2a^2y-5bx-2by\\\\ \\\\&=5a^2x-5bx+2a^2y-2by\\\\ \\\\&=5x(a^2-b)+2y(a^2-b)\\\\ \\\\&=(a^2-b)(5x+2y)\\end{aligned}[\/latex]<\/p>\n<p>Notice that this is the same answer as the previous example with the factors in reverse order.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>Write the following function in factored form: [latex]f(x)=4ax^2+7a^2cx-4bx-7abc[\/latex] by grouping the 1st and 3rd terms, and the 2nd and 4th terms.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm863\">Show Answer<\/span><\/p>\n<div id=\"qhjm863\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex](ax-b)(4x+7ac)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1587\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Factoring a Polynomial Using Grouping. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Examples and Try Its: hjm863, hjm792, hjm969, hjm564, hjm874. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 12: Factoring, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":422608,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Factoring a Polynomial Using Grouping\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 12: Factoring, from Developmental 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University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1587","chapter","type-chapter","status-publish","hentry"],"part":1176,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1587","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/422608"}],"version-history":[{"count":28,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1587\/revisions"}],"predecessor-version":[{"id":4784,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1587\/revisions\/4784"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/1176"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1587\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=1587"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1587"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=1587"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=1587"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}