{"id":837,"date":"2024-04-29T22:01:16","date_gmt":"2024-04-29T22:01:16","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=837"},"modified":"2024-11-20T02:43:27","modified_gmt":"2024-11-20T02:43:27","slug":"factoring-polynomials-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/factoring-polynomials-learn-it-1\/","title":{"raw":"Factoring Polynomials: Learn It 1","rendered":"Factoring Polynomials: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Factor polynomial expressions using the Greatest Common Factor (GCF) and by grouping to simplify expressions.<\/li>\r\n \t<li>Factor trinomials and perfect square trinomials into binomials.<\/li>\r\n \t<li>Break down expressions like differences of squares and cubic equations into their simpler factors.<\/li>\r\n \t<li>Use specific methods to factor expressions that contain fractional or negative exponents.<\/li>\r\n<\/ul>\r\n<\/section>Factoring polynomials is like breaking down a complex mathematical phrase into simpler parts that are easier to handle. Many polynomial expressions, which are combinations of numbers and variables raised to various powers, can often be rewritten in simpler forms by factoring. This process involves finding numbers or expressions that, when multiplied together, produce the original polynomial.\r\n\r\nFactoring is not just a mathematical trick; it's a fundamental tool that helps in solving equations, simplifying expressions, and understanding algebraic structures more deeply.\r\n<h2>Factoring the Greatest Common Factor (GCF)<\/h2>\r\nWhen factoring a polynomial expression, our first step is to check to see if each term contains a common factor. If so, we factor out the greatest amount we can from each term. Finding and factoring out a <strong>greatest common factor <\/strong>from a polynomial is the first skill involved in factoring polynomials.\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">Recall that the <strong>greatest common factor<\/strong> (GCF) of two numbers is the largest number that divides evenly into both numbers. For example, [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].<\/section><section class=\"textbox keyTakeaway\">\r\n<h3>GCF of Polynomials<\/h3>\r\n<div class=\"page\" title=\"Page 82\">\r\n<div class=\"section\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\nThe <strong>greatest common factor (GCF)<\/strong> of polynomials is the largest polynomial that divides evenly into the polynomials.\r\n\r\n&nbsp;\r\n\r\nIdentifying and factoring out the GCF involves recognizing the highest degree of any common variables and the largest factor common to all numerical coefficients.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>To factor out a GCF from a polynomial, first identify the greatest common factor of the terms. You can then use the distributive property \u201cbackwards\u201d to rewrite the polynomial in a factored form.\r\n\r\nDistributive Property: [latex]a\\left(b+c\\right)=ab+ac[\/latex].\r\n\r\nBackward: Let's factor [latex]a[\/latex] out of [latex]ab+ac[\/latex].\r\n<p style=\"text-align: center;\">[latex]ab+ac=a\\left(b+c\\right)[\/latex].<\/p>\r\nWe have seen that we can distribute a factor over a sum or difference. Now we see that we can \"undo\" the distributive property with factoring.\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a polynomial expression, factor out the greatest common factor\r\n<\/strong>\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 \t<li>Determine what the GCF needs to be multiplied by to obtain each term in the expression.<\/li>\r\n \t<li>Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Factor [latex]25b^{3}+10b^{2}[\/latex].<strong>Solution<\/strong>\r\n<ol>\r\n \t<li><strong>Identify the GCF of the Coefficients:\u00a0<\/strong>The coefficients are [latex]25[\/latex] and [latex]10[\/latex]. The GCF of [latex]25[\/latex] and [latex]10[\/latex] is [latex]5[\/latex].<\/li>\r\n \t<li><strong>Identify the GCF of the Variables:<\/strong> The terms include [latex]b^3[\/latex] and [latex]b^2[\/latex]. The GCF of\u00a0[latex]b^3[\/latex] and [latex]b^2[\/latex] is [latex]b^2[\/latex](the term with the lowest power).<\/li>\r\n \t<li><strong>Combine the GCFs:\u00a0<\/strong>Combine the GCF of the coefficients with the GCF of the variables. The overall GCF is [latex]5b^2[\/latex].<\/li>\r\n \t<li><strong>Determine what the GCF needs to be multiplied by to obtain each term in the expression:<\/strong>\r\n<ul>\r\n \t<li>[latex]25b^3 = 5b^2 \\cdot 5b[\/latex]<\/li>\r\n \t<li>[latex]10b^2 = 5b^2 \\cdot 2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Write the Factored Form<\/strong>:<\/li>\r\n<\/ol>\r\n<p style=\"text-align: center;\">[latex]25b^{3}+10b^{2} = 5b^2 \\cdot 5b + 5b^2 \\cdot 2 = 5b^2(5b+2)[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox example\">Factor [latex]6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[\/latex].[reveal-answer q=\"113189\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"113189\"]First find the GCF of the expression. The GCF of [latex]6,45[\/latex], and [latex]21[\/latex] is [latex]3[\/latex]. The GCF of [latex]{x}^{3},{x}^{2}[\/latex], and [latex]x[\/latex] is [latex]x[\/latex]. (Note that the GCF of a set of expressions of the form [latex]{x}^{n}[\/latex] will always be the lowest exponent.) The GCF of [latex]{y}^{3},{y}^{2}[\/latex], and [latex]y[\/latex] is [latex]y[\/latex]. Combine these to find the GCF of the polynomial, [latex]3xy[\/latex].Next, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that [latex]3xy\\left(2{x}^{2}{y}^{2}\\right)=6{x}^{3}{y}^{3}, 3xy\\left(15xy\\right)=45{x}^{2}{y}^{2}[\/latex], and [latex]3xy\\left(7\\right)=21xy[\/latex].\r\n\r\nFinally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by.\r\n<div style=\"text-align: center;\">[latex]\\left(3xy\\right)\\left(2{x}^{2}{y}^{2}+15xy+7\\right)[\/latex]<\/div>\r\n<div>\r\n\r\n<strong>Analysis of the Solution<\/strong>\r\n\r\nAfter factoring, we can check our work by multiplying. Use the distributive property to confirm that [latex]\\left(3xy\\right)\\left(2{x}^{2}{y}^{2}+15xy+7\\right)=6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[\/latex].\r\n\r\n<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/section>The GCF may not always be a monomial. Here is an example of a GCF that is a binomial.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Factor [latex]x\\left({b}^{2}-a\\right)+6\\left({b}^{2}-a\\right)[\/latex] by pulling out the GCF.[reveal-answer q=\"94532\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"94532\"][latex]\\left({b}^{2}-a\\right)\\left(x+6\\right)[\/latex][\/hidden-answer]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18879[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18880[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Factor polynomial expressions using the Greatest Common Factor (GCF) and by grouping to simplify expressions.<\/li>\n<li>Factor trinomials and perfect square trinomials into binomials.<\/li>\n<li>Break down expressions like differences of squares and cubic equations into their simpler factors.<\/li>\n<li>Use specific methods to factor expressions that contain fractional or negative exponents.<\/li>\n<\/ul>\n<\/section>\n<p>Factoring polynomials is like breaking down a complex mathematical phrase into simpler parts that are easier to handle. Many polynomial expressions, which are combinations of numbers and variables raised to various powers, can often be rewritten in simpler forms by factoring. This process involves finding numbers or expressions that, when multiplied together, produce the original polynomial.<\/p>\n<p>Factoring is not just a mathematical trick; it&#8217;s a fundamental tool that helps in solving equations, simplifying expressions, and understanding algebraic structures more deeply.<\/p>\n<h2>Factoring the Greatest Common Factor (GCF)<\/h2>\n<p>When factoring a polynomial expression, our first step is to check to see if each term contains a common factor. If so, we factor out the greatest amount we can from each term. Finding and factoring out a <strong>greatest common factor <\/strong>from a polynomial is the first skill involved in factoring polynomials.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">Recall that the <strong>greatest common factor<\/strong> (GCF) of two numbers is the largest number that divides evenly into both numbers. For example, [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].<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>GCF of Polynomials<\/h3>\n<div class=\"page\" title=\"Page 82\">\n<div class=\"section\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p>The <strong>greatest common factor (GCF)<\/strong> of polynomials is the largest polynomial that divides evenly into the polynomials.<\/p>\n<p>&nbsp;<\/p>\n<p>Identifying and factoring out the GCF involves recognizing the highest degree of any common variables and the largest factor common to all numerical coefficients.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<p>To factor out a GCF from a polynomial, first identify the greatest common factor of the terms. You can then use the distributive property \u201cbackwards\u201d to rewrite the polynomial in a factored form.<\/p>\n<p>Distributive Property: [latex]a\\left(b+c\\right)=ab+ac[\/latex].<\/p>\n<p>Backward: Let&#8217;s factor [latex]a[\/latex] out of [latex]ab+ac[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]ab+ac=a\\left(b+c\\right)[\/latex].<\/p>\n<p>We have seen that we can distribute a factor over a sum or difference. Now we see that we can &#8220;undo&#8221; the distributive property with factoring.<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a polynomial expression, factor out the greatest common factor<br \/>\n<\/strong><\/p>\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<li>Determine what the GCF needs to be multiplied by to obtain each term in the expression.<\/li>\n<li>Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Factor [latex]25b^{3}+10b^{2}[\/latex].<strong>Solution<\/strong><\/p>\n<ol>\n<li><strong>Identify the GCF of the Coefficients:\u00a0<\/strong>The coefficients are [latex]25[\/latex] and [latex]10[\/latex]. The GCF of [latex]25[\/latex] and [latex]10[\/latex] is [latex]5[\/latex].<\/li>\n<li><strong>Identify the GCF of the Variables:<\/strong> The terms include [latex]b^3[\/latex] and [latex]b^2[\/latex]. The GCF of\u00a0[latex]b^3[\/latex] and [latex]b^2[\/latex] is [latex]b^2[\/latex](the term with the lowest power).<\/li>\n<li><strong>Combine the GCFs:\u00a0<\/strong>Combine the GCF of the coefficients with the GCF of the variables. The overall GCF is [latex]5b^2[\/latex].<\/li>\n<li><strong>Determine what the GCF needs to be multiplied by to obtain each term in the expression:<\/strong>\n<ul>\n<li>[latex]25b^3 = 5b^2 \\cdot 5b[\/latex]<\/li>\n<li>[latex]10b^2 = 5b^2 \\cdot 2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><strong>Write the Factored Form<\/strong>:<\/li>\n<\/ol>\n<p style=\"text-align: center;\">[latex]25b^{3}+10b^{2} = 5b^2 \\cdot 5b + 5b^2 \\cdot 2 = 5b^2(5b+2)[\/latex]<\/p>\n<\/section>\n<section class=\"textbox example\">Factor [latex]6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q113189\">Show Solution<\/button><\/p>\n<div id=\"q113189\" class=\"hidden-answer\" style=\"display: none\">First find the GCF of the expression. The GCF of [latex]6,45[\/latex], and [latex]21[\/latex] is [latex]3[\/latex]. The GCF of [latex]{x}^{3},{x}^{2}[\/latex], and [latex]x[\/latex] is [latex]x[\/latex]. (Note that the GCF of a set of expressions of the form [latex]{x}^{n}[\/latex] will always be the lowest exponent.) The GCF of [latex]{y}^{3},{y}^{2}[\/latex], and [latex]y[\/latex] is [latex]y[\/latex]. Combine these to find the GCF of the polynomial, [latex]3xy[\/latex].Next, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that [latex]3xy\\left(2{x}^{2}{y}^{2}\\right)=6{x}^{3}{y}^{3}, 3xy\\left(15xy\\right)=45{x}^{2}{y}^{2}[\/latex], and [latex]3xy\\left(7\\right)=21xy[\/latex].<\/p>\n<p>Finally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by.<\/p>\n<div style=\"text-align: center;\">[latex]\\left(3xy\\right)\\left(2{x}^{2}{y}^{2}+15xy+7\\right)[\/latex]<\/div>\n<div>\n<p><strong>Analysis of the Solution<\/strong><\/p>\n<p>After factoring, we can check our work by multiplying. Use the distributive property to confirm that [latex]\\left(3xy\\right)\\left(2{x}^{2}{y}^{2}+15xy+7\\right)=6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[\/latex].<\/p>\n<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/section>\n<p>The GCF may not always be a monomial. Here is an example of a GCF that is a binomial.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Factor [latex]x\\left({b}^{2}-a\\right)+6\\left({b}^{2}-a\\right)[\/latex] by pulling out the GCF.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q94532\">Show Solution<\/button><\/p>\n<div id=\"q94532\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left({b}^{2}-a\\right)\\left(x+6\\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18879\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18879&theme=lumen&iframe_resize_id=ohm18879&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18880\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18880&theme=lumen&iframe_resize_id=ohm18880&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":11,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":55,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/837"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":23,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/837\/revisions"}],"predecessor-version":[{"id":6226,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/837\/revisions\/6226"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/55"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/837\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=837"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=837"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=837"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=837"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}