{"id":58,"date":"2023-06-21T13:22:28","date_gmt":"2023-06-21T13:22:28","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/greatest-common-factor\/"},"modified":"2023-08-07T01:34:35","modified_gmt":"2023-08-07T01:34:35","slug":"greatest-common-factor","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/greatest-common-factor\/","title":{"raw":"RP2.2   Greatest Common Factor","rendered":"RP2.2   Greatest Common Factor"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the greatest common factor of a list of expressions<\/li>\r\n \t<li>Find the greatest common factor of a polynomial<\/li>\r\n<\/ul>\r\n<\/div>\r\n<strong>Factors<\/strong> are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number. For example, [latex]2[\/latex] and\u00a0[latex]10[\/latex] are factors of\u00a0[latex]20[\/latex], as are\u00a0[latex]4, 5, 1, 20[\/latex]. To factor a number is to rewrite it as a product. [latex]20=4\\cdot{5}[\/latex] or [latex]20=1\\cdot{20}[\/latex]. In algebra, we use the word factor as both a noun \u2013 something being multiplied \u2013 and as a verb \u2013 the action of rewriting a sum or difference as a product.\u00a0<strong>Factoring<\/strong> is very helpful in simplifying expressions and solving equations involving\u00a0polynomials.\r\n\r\nThe <strong>greatest common factor<\/strong> (GCF) of two numbers is the largest number that divides evenly into both numbers: [latex]4[\/latex] is the GCF of [latex]16[\/latex] and [latex]20[\/latex] because it is the largest number that divides evenly into both [latex]16[\/latex] and [latex]20[\/latex]. The GCF of two or more algebraic expressions 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 algebraic expression that divides evenly into both [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex].\r\n<h3>Find the GCF of a list of algebraic expressions<\/h3>\r\nWe begin by finding the GCF of a list of numbers, then we'll extend the technique to monomial expressions containing variables.\r\n\r\nA good technique for finding the GCF of a list of numbers is to write each number as a product of its prime factors. Then, match all the common factors between each prime factorization. The product of all the common factors will build the greatest common factor.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the greatest common factor of [latex]24[\/latex] and [latex]36[\/latex].\r\n\r\n[reveal-answer q=\"863750\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"863750\"]\r\n<table id=\"eip-id1168464918810\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\"Three columns are shown. The top row of the first column says, \">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 194px;\"><strong>Step 1:<\/strong> Factor each coefficient into primes. Write all variables with exponents in expanded form.<\/td>\r\n<td style=\"width: 199.55px;\">Factor [latex]24[\/latex] and [latex]36[\/latex].<\/td>\r\n<td style=\"width: 426.45px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224611\/CNX_BMath_Figure_10_06_024_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 194px;\"><strong>Step 2:<\/strong> List all factors--matching common factors in a column.<\/td>\r\n<td style=\"width: 199.55px;\"><\/td>\r\n<td style=\"width: 426.45px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224614\/CNX_BMath_Figure_10_06_024_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 194px;\">In each column, circle the common factors.<\/td>\r\n<td style=\"width: 199.55px;\">Circle the [latex]2, 2[\/latex], and [latex]3[\/latex] that are shared by both numbers.<\/td>\r\n<td style=\"width: 426.45px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224615\/CNX_BMath_Figure_10_06_024_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 194px;\"><strong>Step 3:<\/strong> Bring down the common factors that all expressions share.<\/td>\r\n<td style=\"width: 199.55px;\">Bring down the [latex]2, 2, 3[\/latex] and then multiply.<\/td>\r\n<td style=\"width: 426.45px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 194px;\"><strong>Step 4:<\/strong> Multiply the factors.<\/td>\r\n<td style=\"width: 199.55px;\"><\/td>\r\n<td style=\"width: 426.45px;\">The GCF of [latex]24[\/latex] and [latex]36[\/latex] is [latex]12[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that since the GCF is a factor of both numbers, [latex]24[\/latex] and [latex]36[\/latex] can be written as multiples of [latex]12[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}24=12\\cdot 2\\\\ 36=12\\cdot 3\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nHere's a summary of the technique.\r\n<div class=\"textbox shaded\">\r\n<h3>Find the greatest common factor<\/h3>\r\n<ol id=\"eip-id1168468531103\" class=\"stepwise\">\r\n \t<li>Factor each coefficient into primes. Write all variables with exponents in expanded form.<\/li>\r\n \t<li>List all factors\u2014matching common factors in a column. In each column, circle the common factors.<\/li>\r\n \t<li>Bring down the common factors that all expressions share.<\/li>\r\n \t<li>Multiply the factors.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146326[\/ohm_question]\r\n\r\n<\/div>\r\nIn the previous example, we found the greatest common factor of a list of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients.\r\n\r\nTo find the GCF of an expression containing variable terms, first find the GCF of the coefficients, then find the GCF of the variables. The GCF of the variables will be the smallest degree of each variable that appears in each term. Here's an example using the matching method from the example above.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the greatest common factor of [latex]5x\\text{ and }15[\/latex].\r\n[reveal-answer q=\"470279\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"470279\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466996785\" class=\"unnumbered unstyled\" summary=\"The left side says, \">\r\n<tbody>\r\n<tr>\r\n<td>Factor each number into primes.\r\n\r\nCircle the common factors in each column.\r\n\r\nBring down the common factors.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224617\/CNX_BMath_Figure_10_06_025_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>The GCF of [latex]5x[\/latex] and [latex]15[\/latex] is [latex]5[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146327[\/ohm_question]\r\n\r\n<\/div>\r\nIn the examples so far, the GCF was a constant. In the next two examples we will get variables in the greatest common factor.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the greatest common factor of [latex]12{x}^{2}[\/latex] and [latex]18{x}^{3}[\/latex].\r\n[reveal-answer q=\"35972\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"35972\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469763176\" class=\"unnumbered unstyled\" summary=\"The left side says, \">\r\n<tbody>\r\n<tr>\r\n<td>Factor each coefficient into primes and write\r\n\r\nthe variables with exponents in expanded form.\r\n\r\nCircle the common factors in each column.\r\n\r\nBring down the common factors.\r\n\r\nMultiply the factors.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224619\/CNX_BMath_Figure_10_06_026_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>The GCF of [latex]12{x}^{2}[\/latex] and [latex]18{x}^{3}[\/latex] is [latex]6{x}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146328[\/ohm_question]\r\n\r\n<\/div>\r\nHere are some examples of finding the GCF of a list of more than two expressions.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the greatest common factor of [latex]14{x}^{3},8{x}^{2},10x[\/latex].\r\n[reveal-answer q=\"215868\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"215868\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469756907\" class=\"unnumbered unstyled\" summary=\"The left side says, \">\r\n<tbody>\r\n<tr>\r\n<td>Factor each coefficient into primes and write\r\n\r\nthe variables with exponents in expanded form.\r\n\r\nCircle the common factors in each column.\r\n\r\nBring down the common factors.\r\n\r\nMultiply the factors.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224620\/CNX_BMath_Figure_10_06_027_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>The GCF of [latex]14{x}^{3}[\/latex] and [latex]8{x}^{2}[\/latex] and [latex]10x[\/latex] is [latex]2x[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146329[\/ohm_question]\r\n\r\n<\/div>\r\nThe common-factor matching method works well for finding the GCF of the coefficients, but when finding the GCF of the variables, you may have noticed that we can simply select the smallest power of a variable that appears in each term.\r\n\r\nIn fact,\u00a0the GCF of a set of expressions in the form [latex]{x}^{n}[\/latex] will always be the exponent of lowest degree.\r\n\r\nWatch the following video to see another example of how to find the GCF of two monomials that have one variable.\r\n\r\nhttps:\/\/youtu.be\/EhkVBXRBC2s\r\n\r\nSometimes you may 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 both contain two variables.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the greatest common factor of [latex]81c^{3}d[\/latex] and [latex]45c^{2}d^{2}[\/latex].\r\n[reveal-answer q=\"930504\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"930504\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,81c^{3}d=3\\cdot3\\cdot3\\cdot3\\cdot{c}\\cdot{c}\\cdot{c}\\cdot{d}\\\\45c^{2}d^{2}=3\\cdot3\\cdot5\\cdot{c}\\cdot{c}\\cdot{d}\\cdot{d}\\\\\\,\\,\\,\\,\\text{GCF}=3\\cdot3\\cdot{c}\\cdot{c}\\cdot{d}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\text{GCF}=9c^{2}d[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe video that follows shows another example of finding the greatest common factor of two monomials with more than one variable.\r\n\r\nhttps:\/\/youtu.be\/GfJvoIO3gKQ\r\n<h3>Find the GCF of a polynomial<\/h3>\r\nNow that you have practiced finding the GCF of a term with one and two variables, the next step is to find the GCF of a polynomial. Later in this module we will apply this idea to factoring\u00a0the GCF out of a polynomial. That is, doing the distributive property \"backwards\" to divide the GCF away from each of the terms in the polynomial. In preparation, practice finding the GCF of a given polynomial.\r\n\r\nRecall that a polynomial is an expression consisting of a sum or difference of terms. To find the GCF of a polynomial, inspect each term for common factors just as you previously did with a list of expressions.\r\n\r\nNo matter how large the polynomial, you can use the same technique described below to identify its GCF.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a polynomial expression, find the greatest common factor.<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Identify the GCF of the coefficients.<\/li>\r\n \t<li>Identify the GCF of the variables.<\/li>\r\n \t<li>Combine to find the GCF of the expression.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the GCF of [latex]6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[\/latex].\r\n\r\n[reveal-answer q=\"572595\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"572595\"]\r\n\r\nThe GCF of [latex]6,45[\/latex], and [latex]21[\/latex] is [latex]3[\/latex].\r\nThe GCF of [latex]{x}^{3},{x}^{2}[\/latex], and [latex]x[\/latex] is [latex]x[\/latex].\r\nAnd the GCF of [latex]{y}^{3},{y}^{2}[\/latex], and [latex]y[\/latex] is [latex]y[\/latex].\r\nCombine these to find the GCF of the polynomial, [latex]3xy[\/latex].\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]14137[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the greatest common factor of a list of expressions<\/li>\n<li>Find the greatest common factor of a polynomial<\/li>\n<\/ul>\n<\/div>\n<p><strong>Factors<\/strong> are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number. For example, [latex]2[\/latex] and\u00a0[latex]10[\/latex] are factors of\u00a0[latex]20[\/latex], as are\u00a0[latex]4, 5, 1, 20[\/latex]. To factor a number is to rewrite it as a product. [latex]20=4\\cdot{5}[\/latex] or [latex]20=1\\cdot{20}[\/latex]. In algebra, we use the word factor as both a noun \u2013 something being multiplied \u2013 and as a verb \u2013 the action of rewriting a sum or difference as a product.\u00a0<strong>Factoring<\/strong> is very helpful in simplifying expressions and solving equations involving\u00a0polynomials.<\/p>\n<p>The <strong>greatest common factor<\/strong> (GCF) of two numbers is the largest number that divides evenly into both numbers: [latex]4[\/latex] is the GCF of [latex]16[\/latex] and [latex]20[\/latex] because it is the largest number that divides evenly into both [latex]16[\/latex] and [latex]20[\/latex]. The GCF of two or more algebraic expressions 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 algebraic expression that divides evenly into both [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex].<\/p>\n<h3>Find the GCF of a list of algebraic expressions<\/h3>\n<p>We begin by finding the GCF of a list of numbers, then we&#8217;ll extend the technique to monomial expressions containing variables.<\/p>\n<p>A good technique for finding the GCF of a list of numbers is to write each number as a product of its prime factors. Then, match all the common factors between each prime factorization. The product of all the common factors will build the greatest common factor.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the greatest common factor of [latex]24[\/latex] and [latex]36[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q863750\">Show Solution<\/span><\/p>\n<div id=\"q863750\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168464918810\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\"Three columns are shown. The top row of the first column says,\">\n<tbody>\n<tr>\n<td style=\"width: 194px;\"><strong>Step 1:<\/strong> Factor each coefficient into primes. Write all variables with exponents in expanded form.<\/td>\n<td style=\"width: 199.55px;\">Factor [latex]24[\/latex] and [latex]36[\/latex].<\/td>\n<td style=\"width: 426.45px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224611\/CNX_BMath_Figure_10_06_024_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 194px;\"><strong>Step 2:<\/strong> List all factors&#8211;matching common factors in a column.<\/td>\n<td style=\"width: 199.55px;\"><\/td>\n<td style=\"width: 426.45px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224614\/CNX_BMath_Figure_10_06_024_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 194px;\">In each column, circle the common factors.<\/td>\n<td style=\"width: 199.55px;\">Circle the [latex]2, 2[\/latex], and [latex]3[\/latex] that are shared by both numbers.<\/td>\n<td style=\"width: 426.45px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224615\/CNX_BMath_Figure_10_06_024_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 194px;\"><strong>Step 3:<\/strong> Bring down the common factors that all expressions share.<\/td>\n<td style=\"width: 199.55px;\">Bring down the [latex]2, 2, 3[\/latex] and then multiply.<\/td>\n<td style=\"width: 426.45px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 194px;\"><strong>Step 4:<\/strong> Multiply the factors.<\/td>\n<td style=\"width: 199.55px;\"><\/td>\n<td style=\"width: 426.45px;\">The GCF of [latex]24[\/latex] and [latex]36[\/latex] is [latex]12[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that since the GCF is a factor of both numbers, [latex]24[\/latex] and [latex]36[\/latex] can be written as multiples of [latex]12[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}24=12\\cdot 2\\\\ 36=12\\cdot 3\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Here&#8217;s a summary of the technique.<\/p>\n<div class=\"textbox shaded\">\n<h3>Find the greatest common factor<\/h3>\n<ol id=\"eip-id1168468531103\" class=\"stepwise\">\n<li>Factor each coefficient into primes. Write all variables with exponents in expanded form.<\/li>\n<li>List all factors\u2014matching common factors in a column. In each column, circle the common factors.<\/li>\n<li>Bring down the common factors that all expressions share.<\/li>\n<li>Multiply the factors.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146326\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146326&theme=oea&iframe_resize_id=ohm146326&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the previous example, we found the greatest common factor of a list of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients.<\/p>\n<p>To find the GCF of an expression containing variable terms, first find the GCF of the coefficients, then find the GCF of the variables. The GCF of the variables will be the smallest degree of each variable that appears in each term. Here&#8217;s an example using the matching method from the example above.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the greatest common factor of [latex]5x\\text{ and }15[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q470279\">Show Solution<\/span><\/p>\n<div id=\"q470279\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466996785\" class=\"unnumbered unstyled\" summary=\"The left side says,\">\n<tbody>\n<tr>\n<td>Factor each number into primes.<\/p>\n<p>Circle the common factors in each column.<\/p>\n<p>Bring down the common factors.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224617\/CNX_BMath_Figure_10_06_025_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>The GCF of [latex]5x[\/latex] and [latex]15[\/latex] is [latex]5[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146327\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146327&theme=oea&iframe_resize_id=ohm146327&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the examples so far, the GCF was a constant. In the next two examples we will get variables in the greatest common factor.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the greatest common factor of [latex]12{x}^{2}[\/latex] and [latex]18{x}^{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q35972\">Show Solution<\/span><\/p>\n<div id=\"q35972\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469763176\" class=\"unnumbered unstyled\" summary=\"The left side says,\">\n<tbody>\n<tr>\n<td>Factor each coefficient into primes and write<\/p>\n<p>the variables with exponents in expanded form.<\/p>\n<p>Circle the common factors in each column.<\/p>\n<p>Bring down the common factors.<\/p>\n<p>Multiply the factors.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224619\/CNX_BMath_Figure_10_06_026_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>The GCF of [latex]12{x}^{2}[\/latex] and [latex]18{x}^{3}[\/latex] is [latex]6{x}^{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146328\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146328&theme=oea&iframe_resize_id=ohm146328&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Here are some examples of finding the GCF of a list of more than two expressions.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the greatest common factor of [latex]14{x}^{3},8{x}^{2},10x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q215868\">Show Solution<\/span><\/p>\n<div id=\"q215868\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469756907\" class=\"unnumbered unstyled\" summary=\"The left side says,\">\n<tbody>\n<tr>\n<td>Factor each coefficient into primes and write<\/p>\n<p>the variables with exponents in expanded form.<\/p>\n<p>Circle the common factors in each column.<\/p>\n<p>Bring down the common factors.<\/p>\n<p>Multiply the factors.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224620\/CNX_BMath_Figure_10_06_027_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>The GCF of [latex]14{x}^{3}[\/latex] and [latex]8{x}^{2}[\/latex] and [latex]10x[\/latex] is [latex]2x[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146329\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146329&theme=oea&iframe_resize_id=ohm146329&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The common-factor matching method works well for finding the GCF of the coefficients, but when finding the GCF of the variables, you may have noticed that we can simply select the smallest power of a variable that appears in each term.<\/p>\n<p>In fact,\u00a0the GCF of a set of expressions in the form [latex]{x}^{n}[\/latex] will always be the exponent of lowest degree.<\/p>\n<p>Watch the following video to see another example of how to find the GCF of two monomials that have one variable.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Determine the GCF of Two Monomials (One Variables)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/EhkVBXRBC2s?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Sometimes you may 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 both contain two variables.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the greatest common factor of [latex]81c^{3}d[\/latex] and [latex]45c^{2}d^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q930504\">Show Solution<\/span><\/p>\n<div id=\"q930504\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,81c^{3}d=3\\cdot3\\cdot3\\cdot3\\cdot{c}\\cdot{c}\\cdot{c}\\cdot{d}\\\\45c^{2}d^{2}=3\\cdot3\\cdot5\\cdot{c}\\cdot{c}\\cdot{d}\\cdot{d}\\\\\\,\\,\\,\\,\\text{GCF}=3\\cdot3\\cdot{c}\\cdot{c}\\cdot{d}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\text{GCF}=9c^{2}d[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The video that follows shows another example of finding the greatest common factor of two monomials with more than one variable.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Determine the GCF of Two Monomials (Two Variables)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/GfJvoIO3gKQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Find the GCF of a polynomial<\/h3>\n<p>Now that you have practiced finding the GCF of a term with one and two variables, the next step is to find the GCF of a polynomial. Later in this module we will apply this idea to factoring\u00a0the GCF out of a polynomial. That is, doing the distributive property &#8220;backwards&#8221; to divide the GCF away from each of the terms in the polynomial. In preparation, practice finding the GCF of a given polynomial.<\/p>\n<p>Recall that a polynomial is an expression consisting of a sum or difference of terms. To find the GCF of a polynomial, inspect each term for common factors just as you previously did with a list of expressions.<\/p>\n<p>No matter how large the polynomial, you can use the same technique described below to identify its GCF.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a polynomial expression, find the greatest common factor.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Identify the GCF of the coefficients.<\/li>\n<li>Identify the GCF of the variables.<\/li>\n<li>Combine to find the GCF of the expression.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the GCF of [latex]6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q572595\">Show Solution<\/span><\/p>\n<div id=\"q572595\" class=\"hidden-answer\" style=\"display: none\">\n<p>The GCF of [latex]6,45[\/latex], and [latex]21[\/latex] is [latex]3[\/latex].<br \/>\nThe GCF of [latex]{x}^{3},{x}^{2}[\/latex], and [latex]x[\/latex] is [latex]x[\/latex].<br \/>\nAnd the GCF of [latex]{y}^{3},{y}^{2}[\/latex], and [latex]y[\/latex] is [latex]y[\/latex].<br \/>\nCombine these to find the GCF of the polynomial, [latex]3xy[\/latex].\n<\/p><\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm14137\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14137&theme=oea&iframe_resize_id=ohm14137&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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-58\">\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>Question ID 146329, 146328, 146327, 146326, 14137. <strong>Authored 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>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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Determine the GCF of Two Monomials (One Variables). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/EhkVBXRBC2s\">https:\/\/youtu.be\/EhkVBXRBC2s<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Determine the GCF of Two Monomials (Two Variables). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/GfJvoIO3gKQ\">https:\/\/youtu.be\/GfJvoIO3gKQ<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/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":395986,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Question ID 146329, 146328, 146327, 146326, 14137\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Determine the GCF of Two Monomials (One Variables)\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/EhkVBXRBC2s\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"cc\",\"description\":\"Ex: Determine the GCF of Two Monomials (Two Variables)\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/GfJvoIO3gKQ\",\"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 Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\" http:\/\/nrocnetwork.org\/dm-opentext\",\"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-58","chapter","type-chapter","status-publish","hentry"],"part":54,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/58","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/users\/395986"}],"version-history":[{"count":4,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/58\/revisions"}],"predecessor-version":[{"id":1262,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/58\/revisions\/1262"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/54"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/58\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=58"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=58"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=58"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=58"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}