{"id":10891,"date":"2017-06-05T21:40:48","date_gmt":"2017-06-05T21:40:48","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10891"},"modified":"2020-10-22T09:31:17","modified_gmt":"2020-10-22T09:31:17","slug":"finding-the-greatest-common-factor-from-two-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/finding-the-greatest-common-factor-from-two-expressions\/","title":{"raw":"13.1.b - Finding the Greatest Common Factor","rendered":"13.1.b &#8211; Finding the 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 multiple numbers<\/li>\r\n \t<li>Find the greatest common factor of monomials<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the section on the zero product principle, we showed that using the techniques for solving equations that we learned for linear equations did not work to solve\r\n<p style=\"text-align: center\">[latex]t\\left(5-t\\right)=0[\/latex]<\/p>\r\n<p style=\"text-align: left\">But because the equation was written as the product of two terms, we could use the zero product principle. What if we are given a polynomial equation that is not written as a product of two terms, such as this one [latex]2y^2+4y=0[\/latex]? We can use a technique called factoring, where we try to find factors that can be divided into each term of the polynomial so it can be rewritten as a product.<\/p>\r\n<p style=\"text-align: left\">In this section we will explore how to find common factors from the terms of a polynomial, and rewrite it as a product. \u00a0This technique will help us\u00a0solve polynomial equations in the next section.<\/p>\r\n\r\n<h2>Finding the Greatest Common Factor of Two Numbers<\/h2>\r\nEarlier we multiplied factors together to get a product.\u00a0\u00a0<strong>Factors<\/strong> are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number: [latex]2[\/latex] and [latex]10[\/latex] are factors of [latex]20[\/latex], as are \u00a0[latex]4[\/latex] and [latex]5[\/latex] and [latex]1[\/latex] and [latex]20[\/latex]. To factor a number is to rewrite it as a product. [latex]20=4\\cdot5[\/latex]. In algebra, we use the word factor as both a noun - something being multiplied - and as a verb - rewriting a sum or difference as a product.\r\n\r\nNow, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224609\/CNX_BMath_Figure_10_06_001_img.png\" alt=\"On the left, the equation 8 times 7 equals 56 is shown. 8 and 7 are labeled factors, 56 is labeled product. On the right, the equation 2x times parentheses x plus 3 equals 2 x squared plus 6x is shown. 2x and x plus 3 are labeled factors, 2 x squared plus 6x is labeled product. There is an arrow on top pointing to the right that says \" \/>\r\n<a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/finding-the-least-common-multiple-of-two-numbers\/\">We also factored numbers to find the least common multiple (LCM) of two or more numbers<\/a>. Now we will factor expressions and find the <em>greatest common factor<\/em> of two or more expressions. The method we use is similar to what we used to find the LCM.\r\n\r\nWhen we studied fractions, we learned that the <strong>greatest common factor<\/strong> (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, [latex]4[\/latex]\u00a0is the GCF of [latex]16[\/latex]\u00a0and [latex]20[\/latex] because it is the largest number that divides evenly into both [latex]16[\/latex]\u00a0and [latex]20[\/latex].The GCF of polynomials works the same way: [latex]4x[\/latex]\u00a0is the GCF of [latex]16x[\/latex]\u00a0and [latex]20x^2[\/latex]\u00a0because it is the largest polynomial that divides evenly into both [latex]16x[\/latex] and [latex]20x^2[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Greatest Common Factor<\/h3>\r\nThe greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.\r\n\r\n<\/div>\r\nFirst we will show how to find the greatest common factor of two numbers.\r\n\r\n[caption id=\"attachment_4759\" align=\"alignleft\" width=\"361\"]<img class=\" wp-image-4759\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/10015359\/Screen-Shot-2016-06-09-at-6.53.22-PM-300x201.png\" alt=\"Prime numbers written with dice including 5, 41, 19, 61, and many others\" width=\"361\" height=\"242\" \/> Prime Numbers[\/caption]\r\n\r\n&nbsp;\r\n\r\nTo get acquainted with the idea of factoring, let\u2019s first find the <strong>greatest common factor (GCF)<\/strong> of two whole numbers. The GCF of two numbers is the greatest number that is a factor of <i>both<\/i> of the numbers. Take the numbers [latex]50[\/latex] and [latex]30[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}50=10\\cdot5\\\\30=10\\cdot3\\end{array}[\/latex]<\/p>\r\nTheir greatest common factor is [latex]10[\/latex], since [latex]10[\/latex] is the greatest factor that both numbers have in common.\r\n\r\nTo find the GCF of greater numbers, you can factor each number to find their prime factors, identify the prime factors they have in common, and then multiply those together.\u00a0\u00a0A <strong>prime factor<\/strong> is similar to a <strong>prime number<\/strong>\u2014it has only itself and 1 as factors. The process of breaking a number down into its prime factors is called <strong>prime factorization<\/strong>.\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\nSolution\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\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the greatest common factor of [latex]210[\/latex] and [latex]168[\/latex].\r\n\r\n[reveal-answer q=\"803757\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"803757\"]\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\,\\,\\,\\,210=2\\cdot3\\cdot5\\cdot7\\\\\\,\\,\\,\\,168=2\\cdot2\\cdot2\\cdot3\\cdot7\\\\\\text{GCF}=2\\cdot3\\cdot7\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\text{GCF}=42[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div>Because the GCF is the product of the prime factors that these numbers have in common, you know that it is a factor of both numbers. (If you want to test this, go ahead and divide both [latex]210[\/latex] and[latex]168[\/latex] by [latex]42[\/latex]\u2014they are both evenly divisible by this number!)<\/div>\r\nThe video that follows shows another example of finding the greatest common factor of two whole numbers.\r\n\r\nhttps:\/\/youtu.be\/KbBJcdDY_VE\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\n<h2>Greatest Common Factor of Polynomials<\/h2>\r\nIn the previous example, we found the greatest common factor of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients.\u00a0 To factor a <strong>polynomial<\/strong>, you rewrite it as a product. Any integer can be written as the product of factors, and we can apply this technique to\u00a0<strong>monomials<\/strong> or polynomials. <strong>Factoring<\/strong> is very helpful in simplifying and solving equations using polynomials.\r\n\r\nFinding the greatest common factor in a set of monomials is not very different from finding the GCF of two whole numbers. The method remains the same: factor each monomial independently, look for common factors, and then multiply them to get the GCF.\u00a0We summarize below a list of steps that can help you to find the greatest common factor.\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&nbsp;\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 greatest common factor 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=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the greatest common factor of [latex]25b^{3}[\/latex] and [latex]10b^{2}[\/latex].\r\n\r\n[reveal-answer q=\"210634\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"210634\"]\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\,\\,25b^{3}=5\\cdot5\\cdot{b}\\cdot{b}\\cdot{b}\\\\\\,\\,10b^{2}=5\\cdot2\\cdot{b}\\cdot{b}\\\\\\text{GCF}=5\\cdot{b}\\cdot{b}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\text{GCF}=5b^{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe monomials have the factors [latex]5[\/latex], <i>b<\/i>, and <i>b<\/i> in common, which means their greatest common factor is [latex]5\\cdot{b}\\cdot{b}[\/latex], or simply [latex]5b^{2}[\/latex].\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\n&nbsp;\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\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\nWe can also factor expressions that have more than one variable.\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\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]39942[\/ohm_question]\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","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the greatest common factor of multiple numbers<\/li>\n<li>Find the greatest common factor of monomials<\/li>\n<\/ul>\n<\/div>\n<p>In the section on the zero product principle, we showed that using the techniques for solving equations that we learned for linear equations did not work to solve<\/p>\n<p style=\"text-align: center\">[latex]t\\left(5-t\\right)=0[\/latex]<\/p>\n<p style=\"text-align: left\">But because the equation was written as the product of two terms, we could use the zero product principle. What if we are given a polynomial equation that is not written as a product of two terms, such as this one [latex]2y^2+4y=0[\/latex]? We can use a technique called factoring, where we try to find factors that can be divided into each term of the polynomial so it can be rewritten as a product.<\/p>\n<p style=\"text-align: left\">In this section we will explore how to find common factors from the terms of a polynomial, and rewrite it as a product. \u00a0This technique will help us\u00a0solve polynomial equations in the next section.<\/p>\n<h2>Finding the Greatest Common Factor of Two Numbers<\/h2>\n<p>Earlier we multiplied factors together to get a product.\u00a0\u00a0<strong>Factors<\/strong> are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number: [latex]2[\/latex] and [latex]10[\/latex] are factors of [latex]20[\/latex], as are \u00a0[latex]4[\/latex] and [latex]5[\/latex] and [latex]1[\/latex] and [latex]20[\/latex]. To factor a number is to rewrite it as a product. [latex]20=4\\cdot5[\/latex]. In algebra, we use the word factor as both a noun &#8211; something being multiplied &#8211; and as a verb &#8211; rewriting a sum or difference as a product.<\/p>\n<p>Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224609\/CNX_BMath_Figure_10_06_001_img.png\" alt=\"On the left, the equation 8 times 7 equals 56 is shown. 8 and 7 are labeled factors, 56 is labeled product. On the right, the equation 2x times parentheses x plus 3 equals 2 x squared plus 6x is shown. 2x and x plus 3 are labeled factors, 2 x squared plus 6x is labeled product. There is an arrow on top pointing to the right that says\" \/><br \/>\n<a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/finding-the-least-common-multiple-of-two-numbers\/\">We also factored numbers to find the least common multiple (LCM) of two or more numbers<\/a>. Now we will factor expressions and find the <em>greatest common factor<\/em> of two or more expressions. The method we use is similar to what we used to find the LCM.<\/p>\n<p>When we studied fractions, we learned that the <strong>greatest common factor<\/strong> (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, [latex]4[\/latex]\u00a0is the GCF of [latex]16[\/latex]\u00a0and [latex]20[\/latex] because it is the largest number that divides evenly into both [latex]16[\/latex]\u00a0and [latex]20[\/latex].The GCF of polynomials works the same way: [latex]4x[\/latex]\u00a0is the GCF of [latex]16x[\/latex]\u00a0and [latex]20x^2[\/latex]\u00a0because it is the largest polynomial that divides evenly into both [latex]16x[\/latex] and [latex]20x^2[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Greatest Common Factor<\/h3>\n<p>The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.<\/p>\n<\/div>\n<p>First we will show how to find the greatest common factor of two numbers.<\/p>\n<div id=\"attachment_4759\" style=\"width: 371px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4759\" class=\"wp-image-4759\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/10015359\/Screen-Shot-2016-06-09-at-6.53.22-PM-300x201.png\" alt=\"Prime numbers written with dice including 5, 41, 19, 61, and many others\" width=\"361\" height=\"242\" \/><\/p>\n<p id=\"caption-attachment-4759\" class=\"wp-caption-text\">Prime Numbers<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>To get acquainted with the idea of factoring, let\u2019s first find the <strong>greatest common factor (GCF)<\/strong> of two whole numbers. The GCF of two numbers is the greatest number that is a factor of <i>both<\/i> of the numbers. Take the numbers [latex]50[\/latex] and [latex]30[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}50=10\\cdot5\\\\30=10\\cdot3\\end{array}[\/latex]<\/p>\n<p>Their greatest common factor is [latex]10[\/latex], since [latex]10[\/latex] is the greatest factor that both numbers have in common.<\/p>\n<p>To find the GCF of greater numbers, you can factor each number to find their prime factors, identify the prime factors they have in common, and then multiply those together.\u00a0\u00a0A <strong>prime factor<\/strong> is similar to a <strong>prime number<\/strong>\u2014it has only itself and 1 as factors. The process of breaking a number down into its prime factors is called <strong>prime factorization<\/strong>.<\/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<p>Solution<\/p>\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 class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the greatest common factor of [latex]210[\/latex] and [latex]168[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q803757\">Show Solution<\/span><\/p>\n<div id=\"q803757\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\,\\,\\,\\,210=2\\cdot3\\cdot5\\cdot7\\\\\\,\\,\\,\\,168=2\\cdot2\\cdot2\\cdot3\\cdot7\\\\\\text{GCF}=2\\cdot3\\cdot7\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\text{GCF}=42[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div>Because the GCF is the product of the prime factors that these numbers have in common, you know that it is a factor of both numbers. (If you want to test this, go ahead and divide both [latex]210[\/latex] and[latex]168[\/latex] by [latex]42[\/latex]\u2014they are both evenly divisible by this number!)<\/div>\n<p>The video that follows shows another example of finding the greatest common factor of two whole numbers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Example:  Determining the Greatest Common Factor\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/KbBJcdDY_VE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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<h2>Greatest Common Factor of Polynomials<\/h2>\n<p>In the previous example, we found the greatest common factor of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients.\u00a0 To factor a <strong>polynomial<\/strong>, you rewrite it as a product. Any integer can be written as the product of factors, and we can apply this technique to\u00a0<strong>monomials<\/strong> or polynomials. <strong>Factoring<\/strong> is very helpful in simplifying and solving equations using polynomials.<\/p>\n<p>Finding the greatest common factor in a set of monomials is not very different from finding the GCF of two whole numbers. The method remains the same: factor each monomial independently, look for common factors, and then multiply them to get the GCF.\u00a0We summarize below a list of steps that can help you to find the greatest common factor.<\/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<p>&nbsp;<\/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 greatest common factor 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=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the greatest common factor of [latex]25b^{3}[\/latex] and [latex]10b^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q210634\">Show Solution<\/span><\/p>\n<div id=\"q210634\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\,\\,25b^{3}=5\\cdot5\\cdot{b}\\cdot{b}\\cdot{b}\\\\\\,\\,10b^{2}=5\\cdot2\\cdot{b}\\cdot{b}\\\\\\text{GCF}=5\\cdot{b}\\cdot{b}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\text{GCF}=5b^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The monomials have the factors [latex]5[\/latex], <i>b<\/i>, and <i>b<\/i> in common, which means their greatest common factor is [latex]5\\cdot{b}\\cdot{b}[\/latex], or simply [latex]5b^{2}[\/latex].<\/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>&nbsp;<\/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>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-2\" 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>We can also factor expressions that have more than one variable.<\/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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm39942\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=39942&theme=oea&iframe_resize_id=ohm39942&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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-3\" 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\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-10891\">\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. <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><\/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><\/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 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