{"id":4769,"date":"2017-06-07T18:56:52","date_gmt":"2017-06-07T18:56:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-greatest-common-factor-of-a-polynomial\/"},"modified":"2017-08-16T01:57:25","modified_gmt":"2017-08-16T01:57:25","slug":"read-greatest-common-factor-of-a-polynomial","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-greatest-common-factor-of-a-polynomial\/","title":{"raw":"Greatest Common Factor of a Polynomial","rendered":"Greatest Common Factor of a Polynomial"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Review the concept of greatest common factor<\/li>\r\n \t<li>Factor a Polynomial<\/li>\r\n<\/ul>\r\n<\/div>\r\n\r\n[caption id=\"attachment_4758\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-4758\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185647\/Screen-Shot-2016-06-09-at-6.49.45-PM-300x277.png\" alt=\"Factors of 54=6 times 9 where 2 times 3 =6 and where 3 times 3 = 9, therefore 54=2 times 3 times 3 times 3\" width=\"300\" height=\"277\" \/> Factor Tree for 54[\/caption]\r\n\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 style=\"text-align: left\">Greatest Common Factor<\/h2>\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, 4\u00a0is the GCF of\u00a016\u00a0and 20 because it is the largest number that divides evenly into both 16\u00a0and 20.The GCF of polynomials works the same way:\u00a04x\u00a0is the GCF of 16x\u00a0and [latex]20x^2[\/latex]\u00a0because it is the largest polynomial that divides evenly into both 16x and [latex]20x^2[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>A General Note: Greatest Common Factor<\/h3>\r\nThe <strong>greatest common factor<\/strong> (GCF) of a group of given polynomials is the largest polynomial that divides evenly into the polynomials.\r\n\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: 2 and 10 are factors of 20, as are 4 and 5 and 1 and 20. 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\nTo 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\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\/2011\/2017\/06\/07185649\/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\nA <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\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 50 and 30.\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 10, since 10 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.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the greatest common factor of 210 and 168.\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\nBecause 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 210 and 168 by 42\u2014they are both evenly divisible by this number!)\r\n\r\nThe video that follows show another example of finding the greatest common factor of two whole numbers.\r\n\r\nhttps:\/\/youtu.be\/KbBJcdDY_VE\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.\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 5, <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\r\nThe video that follows gives another example of finding the greatest common factor of two monomials with only one variable.\r\n\r\nhttps:\/\/youtu.be\/EhkVBXRBC2s\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<h2>Factor a Polynomial<\/h2>\r\n[caption id=\"attachment_4784\" align=\"alignleft\" width=\"300\"]<img class=\"size-medium wp-image-4784\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185651\/Screen-Shot-2016-06-10-at-11.40.44-AM-300x190.png\" alt=\"Beetles pinned to a surface as a collection with a mini volkswagen beetle car in the mix.\" width=\"300\" height=\"190\" \/> One of these things is not like the others.[\/caption]\r\n\r\nBefore we solve polynomial\u00a0equations, we will practice finding the greatest common factor of a polynomial. If you can find common factors for each term of a polynomial, then you can factor it, and solving will be easier.\r\n\r\nTo help you practice finding common factors, identify factors that the terms of the polynomial have in common in the table below.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Polynomial<\/th>\r\n<th>Terms<\/th>\r\n<th>Common Factors<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]6x+9[\/latex]<\/td>\r\n<td>6<em>x<\/em> and 9<\/td>\r\n<td>3 is a factor of 6<i>x<\/i> and 9<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a^{2}\u20132a[\/latex]<\/td>\r\n<td>[latex]a^{2}[\/latex] and [latex]\u22122a[\/latex]<\/td>\r\n<td><i>a<\/i> is a factor of [latex]a^{2}[\/latex] and [latex]\u22122a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4c^{3}+4c[\/latex]<\/td>\r\n<td>[latex]4c^{3}[\/latex] and [latex]4c[\/latex]<\/td>\r\n<td>4 and <i>c<\/i> are factors of [latex]4c^{3}[\/latex] and 4<i>c<\/i><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTo factor a polynomial, first identify the greatest common factor of the terms. You can then use the distributive property to rewrite the polynomial in a factored form. Recall that the <strong>distributive property of multiplication over addition<\/strong> states that a product of a number and a sum is the same as the sum of the products.\r\n<div class=\"textbox shaded\">\r\n<h4>Distributive Property Forward and Backward<\/h4>\r\nForward: Product of a number and a sum: [latex]a\\left(b+c\\right)=a\\cdot{b}+a\\cdot{c}[\/latex]. You can say that \u201c[latex]a[\/latex] is being distributed over [latex]b+c[\/latex].\u201d\r\n\r\nBackward: Sum of the products: [latex]a\\cdot{b}+a\\cdot{c}=a\\left(b+c\\right)[\/latex]. Here you can say that \u201c<em>a<\/em> is being factored out.\u201d\r\n\r\nWe first learned that we could distribute a factor over a sum or difference, now we are learning that we can \"undo\" the distributive property with factoring.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFactor [latex]25b^{3}+10b^{2}[\/latex].\r\n\r\n[reveal-answer q=\"716902\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"716902\"]Find the GCF. From a previous example, you found the GCF of [latex]25b^{3}[\/latex] and [latex]10b^{2}[\/latex] to be [latex]5b^{2}[\/latex].\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}=5b^{2}\\end{array}[\/latex]<\/p>\r\nRewrite each term with the GCF as one factor.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}25b^{3} = 5b^{2}\\cdot5b\\\\10b^{2}=5b^{2}\\cdot2\\end{array}[\/latex]<\/p>\r\nRewrite the polynomial using the factored terms in place of the original terms.\r\n<p style=\"text-align: center\">[latex]5b^{2}\\left(5b\\right)+5b^{2}\\left(2\\right)[\/latex]<\/p>\r\nFactor out the [latex]5b^{2}[\/latex].\r\n<p style=\"text-align: center\">[latex]5b^{2}\\left(5b+2\\right)[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]5b^{2}\\left(5b+2\\right)[\/latex]\r\n\r\n[\/hidden-answer]\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]. You can check this by doing the multiplication. [latex]5b^{2}\\left(5b+2\\right)=25b^{3}+10b^{2}[\/latex].\r\n\r\nNote that if you do not factor the greatest common factor at first, you can continue factoring, rather than start all over.\r\n\r\nFor example:\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}25b^{3}+10b^{2}=5\\left(5b^{3}+2b^{2}\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Factor out }5.\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=5b^{2}\\left(5b+2\\right) \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Factor out }b^{2}.\\end{array}[\/latex]<\/p>\r\nNotice that you arrive at the same simplified form whether you factor out the GCF immediately or if you pull out factors individually.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFactor [latex]81c^{3}d+45c^{2}d^{2}[\/latex].\r\n\r\n[reveal-answer q=\"809701\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"809701\"]Factor [latex]81c^{3}d[\/latex].\r\n<p style=\"text-align: center\">[latex]3\\cdot3\\cdot9\\cdot{c}\\cdot{c}\\cdot{c}\\cdot{d}[\/latex]<\/p>\r\nFactor [latex]45c^{2}d^{2}[\/latex].\r\n<p style=\"text-align: center\">[latex]3\\cdot3\\cdot5\\cdot{c}\\cdot{c}\\cdot{d}\\cdot{d}[\/latex]<\/p>\r\nFind the GCF.\r\n<p style=\"text-align: center\">[latex]3\\cdot3\\cdot{c}\\cdot{c}\\cdot{d}=9c^{2}d[\/latex]<\/p>\r\nRewrite each term as the product of the GCF and the remaining terms.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\,\\,81c^{3}d=9c^{2}d\\left(9c\\right)\\\\45c^{2}d^{2}=9c^{2}d\\left(5d\\right)\\end{array}[\/latex]<\/p>\r\nRewrite the polynomial expression using the factored terms in place of the original terms.\r\n<p style=\"text-align: center\">[latex]9c^{2}d\\left(9c\\right)+9c^{2}d\\left(5d\\right)[\/latex]<\/p>\r\nFactor out [latex]9c^{2}d[\/latex]<i>.<\/i>\r\n<p style=\"text-align: center\">[latex]9c^{2}d\\left(9c+5d\\right)[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]9c^{2}d\\left(9c+5d\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video provides two more examples of finding the greatest common factor of a binomial\r\n\r\nhttps:\/\/youtu.be\/25_f_mVab_4\r\n<h2>Summary<\/h2>\r\nA whole number, monomial, or polynomial can be expressed as a product of factors. You can use some of the same logic that you apply to factoring integers to factoring polynomials. To factor a polynomial, first identify the greatest common factor of the terms, and then apply the distributive property to rewrite the expression. Once a polynomial in [latex]a\\cdot{b}+a\\cdot{c}[\/latex] form has been rewritten as [latex]a\\left(b+c\\right)[\/latex], where <i>a<\/i> is the GCF, the polynomial is in factored form.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Review the concept of greatest common factor<\/li>\n<li>Factor a Polynomial<\/li>\n<\/ul>\n<\/div>\n<div id=\"attachment_4758\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4758\" class=\"size-medium wp-image-4758\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185647\/Screen-Shot-2016-06-09-at-6.49.45-PM-300x277.png\" alt=\"Factors of 54=6 times 9 where 2 times 3 =6 and where 3 times 3 = 9, therefore 54=2 times 3 times 3 times 3\" width=\"300\" height=\"277\" \/><\/p>\n<p id=\"caption-attachment-4758\" class=\"wp-caption-text\">Factor Tree for 54<\/p>\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 style=\"text-align: left\">Greatest Common Factor<\/h2>\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, 4\u00a0is the GCF of\u00a016\u00a0and 20 because it is the largest number that divides evenly into both 16\u00a0and 20.The GCF of polynomials works the same way:\u00a04x\u00a0is the GCF of 16x\u00a0and [latex]20x^2[\/latex]\u00a0because it is the largest polynomial that divides evenly into both 16x and [latex]20x^2[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>A General Note: Greatest Common Factor<\/h3>\n<p>The <strong>greatest common factor<\/strong> (GCF) of a group of given polynomials is the largest polynomial that divides evenly into the polynomials.<\/p>\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: 2 and 10 are factors of 20, as are 4 and 5 and 1 and 20. 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>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<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\/2011\/2017\/06\/07185649\/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>A <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<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 50 and 30.<\/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 10, since 10 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.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the greatest common factor of 210 and 168.<\/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<p>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 210 and 168 by 42\u2014they are both evenly divisible by this number!)<\/p>\n<p>The video that follows show 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<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.<\/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 5, <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<p>The video that follows gives another example of finding the greatest common factor of two monomials with only 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<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-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<h2>Factor a Polynomial<\/h2>\n<div id=\"attachment_4784\" style=\"width: 310px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4784\" class=\"size-medium wp-image-4784\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185651\/Screen-Shot-2016-06-10-at-11.40.44-AM-300x190.png\" alt=\"Beetles pinned to a surface as a collection with a mini volkswagen beetle car in the mix.\" width=\"300\" height=\"190\" \/><\/p>\n<p id=\"caption-attachment-4784\" class=\"wp-caption-text\">One of these things is not like the others.<\/p>\n<\/div>\n<p>Before we solve polynomial\u00a0equations, we will practice finding the greatest common factor of a polynomial. If you can find common factors for each term of a polynomial, then you can factor it, and solving will be easier.<\/p>\n<p>To help you practice finding common factors, identify factors that the terms of the polynomial have in common in the table below.<\/p>\n<table>\n<thead>\n<tr>\n<th>Polynomial<\/th>\n<th>Terms<\/th>\n<th>Common Factors<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]6x+9[\/latex]<\/td>\n<td>6<em>x<\/em> and 9<\/td>\n<td>3 is a factor of 6<i>x<\/i> and 9<\/td>\n<\/tr>\n<tr>\n<td>[latex]a^{2}\u20132a[\/latex]<\/td>\n<td>[latex]a^{2}[\/latex] and [latex]\u22122a[\/latex]<\/td>\n<td><i>a<\/i> is a factor of [latex]a^{2}[\/latex] and [latex]\u22122a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4c^{3}+4c[\/latex]<\/td>\n<td>[latex]4c^{3}[\/latex] and [latex]4c[\/latex]<\/td>\n<td>4 and <i>c<\/i> are factors of [latex]4c^{3}[\/latex] and 4<i>c<\/i><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>To factor a polynomial, first identify the greatest common factor of the terms. You can then use the distributive property to rewrite the polynomial in a factored form. Recall that the <strong>distributive property of multiplication over addition<\/strong> states that a product of a number and a sum is the same as the sum of the products.<\/p>\n<div class=\"textbox shaded\">\n<h4>Distributive Property Forward and Backward<\/h4>\n<p>Forward: Product of a number and a sum: [latex]a\\left(b+c\\right)=a\\cdot{b}+a\\cdot{c}[\/latex]. You can say that \u201c[latex]a[\/latex] is being distributed over [latex]b+c[\/latex].\u201d<\/p>\n<p>Backward: Sum of the products: [latex]a\\cdot{b}+a\\cdot{c}=a\\left(b+c\\right)[\/latex]. Here you can say that \u201c<em>a<\/em> is being factored out.\u201d<\/p>\n<p>We first learned that we could distribute a factor over a sum or difference, now we are learning that we can &#8220;undo&#8221; the distributive property with factoring.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Factor [latex]25b^{3}+10b^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q716902\">Show Solution<\/span><\/p>\n<div id=\"q716902\" class=\"hidden-answer\" style=\"display: none\">Find the GCF. From a previous example, you found the GCF of [latex]25b^{3}[\/latex] and [latex]10b^{2}[\/latex] to be [latex]5b^{2}[\/latex].<\/p>\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}=5b^{2}\\end{array}[\/latex]<\/p>\n<p>Rewrite each term with the GCF as one factor.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}25b^{3} = 5b^{2}\\cdot5b\\\\10b^{2}=5b^{2}\\cdot2\\end{array}[\/latex]<\/p>\n<p>Rewrite the polynomial using the factored terms in place of the original terms.<\/p>\n<p style=\"text-align: center\">[latex]5b^{2}\\left(5b\\right)+5b^{2}\\left(2\\right)[\/latex]<\/p>\n<p>Factor out the [latex]5b^{2}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]5b^{2}\\left(5b+2\\right)[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]5b^{2}\\left(5b+2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\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]. You can check this by doing the multiplication. [latex]5b^{2}\\left(5b+2\\right)=25b^{3}+10b^{2}[\/latex].<\/p>\n<p>Note that if you do not factor the greatest common factor at first, you can continue factoring, rather than start all over.<\/p>\n<p>For example:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}25b^{3}+10b^{2}=5\\left(5b^{3}+2b^{2}\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Factor out }5.\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=5b^{2}\\left(5b+2\\right) \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Factor out }b^{2}.\\end{array}[\/latex]<\/p>\n<p>Notice that you arrive at the same simplified form whether you factor out the GCF immediately or if you pull out factors individually.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Factor [latex]81c^{3}d+45c^{2}d^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q809701\">Show Solution<\/span><\/p>\n<div id=\"q809701\" class=\"hidden-answer\" style=\"display: none\">Factor [latex]81c^{3}d[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]3\\cdot3\\cdot9\\cdot{c}\\cdot{c}\\cdot{c}\\cdot{d}[\/latex]<\/p>\n<p>Factor [latex]45c^{2}d^{2}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]3\\cdot3\\cdot5\\cdot{c}\\cdot{c}\\cdot{d}\\cdot{d}[\/latex]<\/p>\n<p>Find the GCF.<\/p>\n<p style=\"text-align: center\">[latex]3\\cdot3\\cdot{c}\\cdot{c}\\cdot{d}=9c^{2}d[\/latex]<\/p>\n<p>Rewrite each term as the product of the GCF and the remaining terms.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\,\\,81c^{3}d=9c^{2}d\\left(9c\\right)\\\\45c^{2}d^{2}=9c^{2}d\\left(5d\\right)\\end{array}[\/latex]<\/p>\n<p>Rewrite the polynomial expression using the factored terms in place of the original terms.<\/p>\n<p style=\"text-align: center\">[latex]9c^{2}d\\left(9c\\right)+9c^{2}d\\left(5d\\right)[\/latex]<\/p>\n<p>Factor out [latex]9c^{2}d[\/latex]<i>.<\/i><\/p>\n<p style=\"text-align: center\">[latex]9c^{2}d\\left(9c+5d\\right)[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]9c^{2}d\\left(9c+5d\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video provides two more examples of finding the greatest common factor of a binomial<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex 1:  Identify GCF and Factor a Binomial\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/25_f_mVab_4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>A whole number, monomial, or polynomial can be expressed as a product of factors. You can use some of the same logic that you apply to factoring integers to factoring polynomials. To factor a polynomial, first identify the greatest common factor of the terms, and then apply the distributive property to rewrite the expression. Once a polynomial in [latex]a\\cdot{b}+a\\cdot{c}[\/latex] form has been rewritten as [latex]a\\left(b+c\\right)[\/latex], where <i>a<\/i> is the GCF, the polynomial is in factored form.<\/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-4769\">\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>Screenshot One of These Things. <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>Screenshot Factor Tree. <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>Screenshot Prime Numbers. <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>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>Example: Determining the Greatest Common Factor. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/KbBJcdDY_VE\">https:\/\/youtu.be\/KbBJcdDY_VE<\/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 (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>Ex 1: Identify GCF and Factor a Binomial. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/25_f_mVab_4\">https:\/\/youtu.be\/25_f_mVab_4<\/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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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