{"id":2215,"date":"2024-07-12T19:04:07","date_gmt":"2024-07-12T19:04:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2215"},"modified":"2025-01-22T06:05:05","modified_gmt":"2025-01-22T06:05:05","slug":"module-12-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/module-12-background-youll-need-2\/","title":{"raw":"Exponential and Logarithmic Equations and Models: Background You'll Need 2","rendered":"Exponential and Logarithmic Equations and Models: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Factor and simplify polynomials and rational expressions&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:513,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0}\">Factor and simplify polynomials and rational expressions<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Factor Polynomials<\/h2>\r\nFactoring is central to simplifying expressions, solving equations, and understanding polynomial behavior. Factoring involves breaking down expressions into simpler, constituent parts. A key step in this process is identifying the greatest common factor (GCF), which simplifies polynomials by dividing out commonalities and reducing complexity.\r\n<h3>Greatest Common Factor<\/h3>\r\nThe <strong>greatest common factor<\/strong> (GCF) of two numbers is the largest number that divides evenly into both numbers.\r\n\r\n<section class=\"textbox 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>The GCF of polynomials works the same way.\r\n\r\n<section class=\"textbox example\">[latex]4x[\/latex] is the GCF of [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex] because it is the largest polynomial that divides evenly into both [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex].<\/section>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.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>greatest common factor (GCF) of a polynomial<\/h3>\r\nThe <strong>greatest common factor<\/strong> (GCF) of a polynomial is the largest polynomial that divides evenly into each term of the polynomial.\r\n\r\n<\/section><section class=\"textbox proTip\">To make it less challenging to find this GCF of the polynomial terms, first look for the GCF of the coefficients, and then look for the GCF of the variables.<\/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 \"backwards\" to rewrite the polynomial in a factored form.\r\n\r\n<section class=\"textbox recall\">The distributive property allows us to multiply a number by a sum or difference inside parentheses and add or subtract the results. Conversely, when we see a common factor shared by all terms, we can factor it out, effectively reversing the distributive process.\r\n<ul>\r\n \t<li>Using the distributive property: [latex]a\\left(b+c\\right)=ab+ac[\/latex].<\/li>\r\n \t<li>Factoring out a common factor: [latex]ab+ac=a\\left(b+c\\right)[\/latex].<\/li>\r\n<\/ul>\r\nThis principle shows us that multiplication distributed across a sum can be \"undone\" through factoring, revealing the GCF and the remaining terms of the polynomial.\r\n\r\n<\/section><section class=\"textbox questionHelp\"><strong>How To: Given a Polynomial Expression, Factor Out the Greatest Common Factor<\/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]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:<center>[latex]\\begin{array}{c} 3xy(2x^2y^2) = 6x^3y^3, \\\\ 3xy(15xy) = 45x^2y^2, \\\\ 3xy(7) = 21xy \\end{array}[\/latex]<\/center>Finally, 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\nAfter factoring, we can check our work by multiplying. Use the distributive property to confirm that\r\n\r\n<center>[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]<\/center>[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\">Watch this video to see more examples of how to factor the GCF from a trinomial.\r\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hbacgfca-3f1RFTIw2Ng\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/3f1RFTIw2Ng?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hbacgfca-3f1RFTIw2Ng\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=6454717&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hbacgfca-3f1RFTIw2Ng&vembed=0&video_id=3f1RFTIw2Ng&video_target=tpm-plugin-hbacgfca-3f1RFTIw2Ng'><\/script><\/p>\r\nYou can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+2+-+Identify+GCF+and+Factor+a+Trinomial_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \"Ex 2: Identify GCF and Factor a Trinomial\" here (opens in new window)<\/a>.\r\n\r\n<\/section>\r\n<h3><strong>Factoring Quadratic Trinomials with a Leading Coefficient of [latex]1[\/latex]<\/strong><\/h3>\r\nWhen factoring polynomials, starting with the greatest common factor (GCF) is standard. However, the GCF is not always the key to simplification, particularly for polynomials without a common factor.\r\n\r\nFor instance, the quadratic [pb_glossary id=\"7006\"]trinomial[\/pb_glossary] [latex]{x}^{2}+5x+6[\/latex] has a GCF of [latex]1[\/latex], but it can be written as the product of the factors [latex]\\left(x+2\\right)[\/latex] and [latex]\\left(x+3\\right)[\/latex].\r\n\r\nTo factor trinomials like [latex]{x}^{2}+bx+c[\/latex], find two numbers that multiply to [latex]c[\/latex] and add up to [latex]b[\/latex].\r\n\r\n<section class=\"textbox example\">The trinomial [latex]{x}^{2}+10x+16[\/latex] can be factored using the numbers [latex]2[\/latex] and [latex]8[\/latex], because [latex]2 \\times 8 =16[\/latex] and [latex]2 + 8 = 10[\/latex]. The trinomial can be rewritten as the product of [latex]\\left(x+2\\right)[\/latex] and [latex]\\left(x+8\\right)[\/latex].<\/section><section class=\"textbox keyTakeaway\">\r\n<h3><strong>factoring quadratic trinomials with a leading coefficient of [latex]1[\/latex]<\/strong><\/h3>\r\nA trinomial of the form [latex]{x}^{2}+bx+c[\/latex] can be written in factored form [latex]\\left(x+p\\right)\\left(x+q\\right)[\/latex] where [latex]p \\times q=c[\/latex] and [latex]p+q=b[\/latex].\r\n\r\n<\/section><section class=\"textbox proTip\">It's a common misconception that all trinomials can be broken down into binomial factors, but this isn't always the case. While many polynomials can be factored in this way, revealing a product of simpler binomials, there are instances where a trinomial is prime and cannot be factored further using real numbers<\/section><section class=\"textbox questionHelp\"><strong>How To: Factoring a Trinomial of the Form [latex]{x}^{2}+bx+c[\/latex]<\/strong>\r\n<ol>\r\n \t<li>Identify all factor pairs of [latex]c[\/latex].<\/li>\r\n \t<li>Find the factor pair where the sum equals [latex]b[\/latex].<\/li>\r\n \t<li>Write the trinomial as the product of two binomials, [latex]\\left(x+p\\right)\\left(x+q\\right)[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>To verify the accuracy of our factorization, we can employ the FOIL method, which stands for First, Outer, Inner, Last. This technique allows us to multiply two binomials and ensures that our factorization is correct. If the expanded expression matches the original polynomial, our factorization is verified.\r\n\r\n<section class=\"textbox recall\">The FOIL method is a process used to multiply two binomials. The acronym FOIL stands for:\r\n<ul>\r\n \t<li><strong>First:<\/strong> Multiply the first terms in each binomial.<\/li>\r\n \t<li><strong>Outer:<\/strong> Multiply the outermost terms in the product.<\/li>\r\n \t<li><strong>Inner:<\/strong> Multiply the innermost terms.<\/li>\r\n \t<li><strong>Last:<\/strong> Multiply the last terms in each binomial.<\/li>\r\n<\/ul>\r\nAfter applying the FOIL method, combine like terms to get the final expanded expression.\r\n\r\n<\/section><section class=\"textbox example\">Factor [latex]{x}^{2}+2x - 15[\/latex].[reveal-answer q=\"88306\"]Show Solution[\/reveal-answer] [hidden-answer a=\"88306\"] We have a trinomial with leading coefficient [latex]1,b=2[\/latex], and [latex]c=-15[\/latex].We need to find two numbers with a product of [latex]-15[\/latex] and a sum of [latex]2[\/latex].\r\n[latex]\\\\[\/latex]\r\nIn the table, we list factors until we find a pair with the desired sum.\r\n<table summary=\"A table with five rows and two columns. The first row has columns labeled: Factors of -15 and Sum of Factors. The entries in the first column are: 1, -15; -1, 15; 3, -5; and -3,5. The entries in the second column are: -14, 14, -2, and 2.\">\r\n<thead>\r\n<tr>\r\n<th>Factors of [latex]-15[\/latex]<\/th>\r\n<th>Sum of Factors<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]1,-15[\/latex]<\/td>\r\n<td>[latex]-14[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1,15[\/latex]<\/td>\r\n<td>[latex]14[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3,-5[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3,5[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow that we have identified [latex]p[\/latex] and [latex]q[\/latex] as [latex]-3[\/latex] and [latex]5[\/latex], write the factored form as [latex]\\left(x - 3\\right)\\left(x+5\\right)[\/latex].\r\n\r\nWe can check our work by multiplying.\r\n\r\nUse FOIL to confirm that [latex]\\left(x - 3\\right)\\left(x+5\\right)={x}^{2}+2x - 15[\/latex]. [\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]288273[\/ohm_question]<\/section>\r\n<h2>Simplify Rational Expressions<\/h2>\r\nA rational expression is formed by dividing one polynomial by another. To simplify these expressions, we use fraction properties, particularly focusing on reducing common factors between the numerator and denominator.\r\n\r\n<section class=\"textbox example\">Here\u2019s the process:\r\n<p style=\"text-align: center;\">[latex]\\text{Original Expression: }\\frac{{x}^{2}+8x+16}{{x}^{2}+11x+28}[\/latex]<\/p>\r\n<strong>Step 1: Factor both the numerator and denominator.<\/strong>\r\n\r\n<center>[latex]\\text{Factorized Form: }\\frac{{(x+4)}{(x+4)}}{\\left(x+4\\right)\\left(x+7\\right)}[\/latex]<\/center>\r\n<div><strong>Step 2: Cancel out common factors.<\/strong><center>[latex]\\text{Simplified Expression: }\\frac{x+4}{x+7}[\/latex].<\/center>By removing the common factor of [latex]x+4[\/latex], we've simplified the rational expression to its reduced form.<\/div>\r\n<\/section><section class=\"textbox questionHelp\"><strong>How To: Simplify a Rational Expression<\/strong>\r\n<ol>\r\n \t<li><strong>Identify the Polynomials<\/strong>: Recognize the numerator and denominator as separate polynomials.<\/li>\r\n \t<li><strong>Factor Completely<\/strong>: Break down both the numerator and the denominator into their prime factors.<\/li>\r\n \t<li><strong>Cancel Common Factors<\/strong>: Look for and cancel out any factors that appear in both the numerator and the denominator.<\/li>\r\n \t<li><strong>Write the Simplified Expression<\/strong>: After canceling the common factors, write down what remains.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Simplify [latex]\\frac{{x}^{2}-9}{{x}^{2}+4x+3}[\/latex].[reveal-answer q=\"568949\"]Show Solution[\/reveal-answer] [hidden-answer a=\"568949\"]<center>[latex]\\begin{array}{lllllllll}\\frac{\\left(x+3\\right)\\left(x - 3\\right)}{\\left(x+3\\right)\\left(x+1\\right)}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Factor the numerator and the denominator}.\\hfill \\\\ \\frac{x - 3}{x+1}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Cancel common factor }\\left(x+3\\right).\\hfill \\end{array}[\/latex]<\/center>We can cancel the common factor because any expression divided by itself is equal to [latex]1[\/latex]. [\/hidden-answer]<\/section><section class=\"textbox questionHelp\"><strong><strong>Can the [latex]{x}^{2}[\/latex] term be cancelled in the above example?\r\n<\/strong><\/strong>\r\n\r\n<hr \/>\r\n\r\nNo. A factor is an expression that is multiplied by another expression. The [latex]{x}^{2}[\/latex] term is not a factor of the numerator or the denominator.\r\n\r\n<\/section><section class=\"textbox example\">Simplify [latex]\\frac{x - 6}{{x}^{2}-36}[\/latex]. [reveal-answer q=\"17752\"]Show Solution[\/reveal-answer] [hidden-answer a=\"17752\"] [latex]\\frac{1}{x+6}[\/latex][\/hidden-answer]<\/section>\r\n<div><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]110917[\/ohm_question]<\/section><\/div>\r\n<section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]110916[\/ohm_question]<\/section><\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Factor and simplify polynomials and rational expressions&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:513,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0}\">Factor and simplify polynomials and rational expressions<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Factor Polynomials<\/h2>\n<p>Factoring is central to simplifying expressions, solving equations, and understanding polynomial behavior. Factoring involves breaking down expressions into simpler, constituent parts. A key step in this process is identifying the greatest common factor (GCF), which simplifies polynomials by dividing out commonalities and reducing complexity.<\/p>\n<h3>Greatest Common Factor<\/h3>\n<p>The <strong>greatest common factor<\/strong> (GCF) of two numbers is the largest number that divides evenly into both numbers.<\/p>\n<section class=\"textbox 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<p>The GCF of polynomials works the same way.<\/p>\n<section class=\"textbox example\">[latex]4x[\/latex] is the GCF of [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex] because it is the largest polynomial that divides evenly into both [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex].<\/section>\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.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>greatest common factor (GCF) of a polynomial<\/h3>\n<p>The <strong>greatest common factor<\/strong> (GCF) of a polynomial is the largest polynomial that divides evenly into each term of the polynomial.<\/p>\n<\/section>\n<section class=\"textbox proTip\">To make it less challenging to find this GCF of the polynomial terms, first look for the GCF of the coefficients, and then look for the GCF of the variables.<\/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 &#8220;backwards&#8221; to rewrite the polynomial in a factored form.<\/p>\n<section class=\"textbox recall\">The distributive property allows us to multiply a number by a sum or difference inside parentheses and add or subtract the results. Conversely, when we see a common factor shared by all terms, we can factor it out, effectively reversing the distributive process.<\/p>\n<ul>\n<li>Using the distributive property: [latex]a\\left(b+c\\right)=ab+ac[\/latex].<\/li>\n<li>Factoring out a common factor: [latex]ab+ac=a\\left(b+c\\right)[\/latex].<\/li>\n<\/ul>\n<p>This principle shows us that multiplication distributed across a sum can be &#8220;undone&#8221; through factoring, revealing the GCF and the remaining terms of the polynomial.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\"><strong>How To: Given a Polynomial Expression, Factor Out the Greatest Common Factor<\/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]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:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{c} 3xy(2x^2y^2) = 6x^3y^3, \\\\ 3xy(15xy) = 45x^2y^2, \\\\ 3xy(7) = 21xy \\end{array}[\/latex]<\/div>\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<p>After factoring, we can check our work by multiplying. Use the distributive property to confirm that<\/p>\n<div style=\"text-align: center;\">[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]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\">Watch this video to see more examples of how to factor the GCF from a trinomial.<br \/>\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hbacgfca-3f1RFTIw2Ng\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/3f1RFTIw2Ng?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hbacgfca-3f1RFTIw2Ng\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=6454717&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-hbacgfca-3f1RFTIw2Ng&#38;vembed=0&#38;video_id=3f1RFTIw2Ng&#38;video_target=tpm-plugin-hbacgfca-3f1RFTIw2Ng\"><\/script><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+2+-+Identify+GCF+and+Factor+a+Trinomial_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for &#8220;Ex 2: Identify GCF and Factor a Trinomial&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n<h3><strong>Factoring Quadratic Trinomials with a Leading Coefficient of [latex]1[\/latex]<\/strong><\/h3>\n<p>When factoring polynomials, starting with the greatest common factor (GCF) is standard. However, the GCF is not always the key to simplification, particularly for polynomials without a common factor.<\/p>\n<p>For instance, the quadratic <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_2215_7006\">trinomial<\/a> [latex]{x}^{2}+5x+6[\/latex] has a GCF of [latex]1[\/latex], but it can be written as the product of the factors [latex]\\left(x+2\\right)[\/latex] and [latex]\\left(x+3\\right)[\/latex].<\/p>\n<p>To factor trinomials like [latex]{x}^{2}+bx+c[\/latex], find two numbers that multiply to [latex]c[\/latex] and add up to [latex]b[\/latex].<\/p>\n<section class=\"textbox example\">The trinomial [latex]{x}^{2}+10x+16[\/latex] can be factored using the numbers [latex]2[\/latex] and [latex]8[\/latex], because [latex]2 \\times 8 =16[\/latex] and [latex]2 + 8 = 10[\/latex]. The trinomial can be rewritten as the product of [latex]\\left(x+2\\right)[\/latex] and [latex]\\left(x+8\\right)[\/latex].<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3><strong>factoring quadratic trinomials with a leading coefficient of [latex]1[\/latex]<\/strong><\/h3>\n<p>A trinomial of the form [latex]{x}^{2}+bx+c[\/latex] can be written in factored form [latex]\\left(x+p\\right)\\left(x+q\\right)[\/latex] where [latex]p \\times q=c[\/latex] and [latex]p+q=b[\/latex].<\/p>\n<\/section>\n<section class=\"textbox proTip\">It&#8217;s a common misconception that all trinomials can be broken down into binomial factors, but this isn&#8217;t always the case. While many polynomials can be factored in this way, revealing a product of simpler binomials, there are instances where a trinomial is prime and cannot be factored further using real numbers<\/section>\n<section class=\"textbox questionHelp\"><strong>How To: Factoring a Trinomial of the Form [latex]{x}^{2}+bx+c[\/latex]<\/strong><\/p>\n<ol>\n<li>Identify all factor pairs of [latex]c[\/latex].<\/li>\n<li>Find the factor pair where the sum equals [latex]b[\/latex].<\/li>\n<li>Write the trinomial as the product of two binomials, [latex]\\left(x+p\\right)\\left(x+q\\right)[\/latex].<\/li>\n<\/ol>\n<\/section>\n<p>To verify the accuracy of our factorization, we can employ the FOIL method, which stands for First, Outer, Inner, Last. This technique allows us to multiply two binomials and ensures that our factorization is correct. If the expanded expression matches the original polynomial, our factorization is verified.<\/p>\n<section class=\"textbox recall\">The FOIL method is a process used to multiply two binomials. The acronym FOIL stands for:<\/p>\n<ul>\n<li><strong>First:<\/strong> Multiply the first terms in each binomial.<\/li>\n<li><strong>Outer:<\/strong> Multiply the outermost terms in the product.<\/li>\n<li><strong>Inner:<\/strong> Multiply the innermost terms.<\/li>\n<li><strong>Last:<\/strong> Multiply the last terms in each binomial.<\/li>\n<\/ul>\n<p>After applying the FOIL method, combine like terms to get the final expanded expression.<\/p>\n<\/section>\n<section class=\"textbox example\">Factor [latex]{x}^{2}+2x - 15[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q88306\">Show Solution<\/button> <\/p>\n<div id=\"q88306\" class=\"hidden-answer\" style=\"display: none\"> We have a trinomial with leading coefficient [latex]1,b=2[\/latex], and [latex]c=-15[\/latex].We need to find two numbers with a product of [latex]-15[\/latex] and a sum of [latex]2[\/latex].<br \/>\n[latex]\\\\[\/latex]<br \/>\nIn the table, we list factors until we find a pair with the desired sum.<\/p>\n<table summary=\"A table with five rows and two columns. The first row has columns labeled: Factors of -15 and Sum of Factors. The entries in the first column are: 1, -15; -1, 15; 3, -5; and -3,5. The entries in the second column are: -14, 14, -2, and 2.\">\n<thead>\n<tr>\n<th>Factors of [latex]-15[\/latex]<\/th>\n<th>Sum of Factors<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]1,-15[\/latex]<\/td>\n<td>[latex]-14[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1,15[\/latex]<\/td>\n<td>[latex]14[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3,-5[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3,5[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now that we have identified [latex]p[\/latex] and [latex]q[\/latex] as [latex]-3[\/latex] and [latex]5[\/latex], write the factored form as [latex]\\left(x - 3\\right)\\left(x+5\\right)[\/latex].<\/p>\n<p>We can check our work by multiplying.<\/p>\n<p>Use FOIL to confirm that [latex]\\left(x - 3\\right)\\left(x+5\\right)={x}^{2}+2x - 15[\/latex]. <\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm288273\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288273&theme=lumen&iframe_resize_id=ohm288273&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Simplify Rational Expressions<\/h2>\n<p>A rational expression is formed by dividing one polynomial by another. To simplify these expressions, we use fraction properties, particularly focusing on reducing common factors between the numerator and denominator.<\/p>\n<section class=\"textbox example\">Here\u2019s the process:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{Original Expression: }\\frac{{x}^{2}+8x+16}{{x}^{2}+11x+28}[\/latex]<\/p>\n<p><strong>Step 1: Factor both the numerator and denominator.<\/strong><\/p>\n<div style=\"text-align: center;\">[latex]\\text{Factorized Form: }\\frac{{(x+4)}{(x+4)}}{\\left(x+4\\right)\\left(x+7\\right)}[\/latex]<\/div>\n<div><strong>Step 2: Cancel out common factors.<\/strong><\/p>\n<div style=\"text-align: center;\">[latex]\\text{Simplified Expression: }\\frac{x+4}{x+7}[\/latex].<\/div>\n<p>By removing the common factor of [latex]x+4[\/latex], we&#8217;ve simplified the rational expression to its reduced form.<\/p><\/div>\n<\/section>\n<section class=\"textbox questionHelp\"><strong>How To: Simplify a Rational Expression<\/strong><\/p>\n<ol>\n<li><strong>Identify the Polynomials<\/strong>: Recognize the numerator and denominator as separate polynomials.<\/li>\n<li><strong>Factor Completely<\/strong>: Break down both the numerator and the denominator into their prime factors.<\/li>\n<li><strong>Cancel Common Factors<\/strong>: Look for and cancel out any factors that appear in both the numerator and the denominator.<\/li>\n<li><strong>Write the Simplified Expression<\/strong>: After canceling the common factors, write down what remains.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Simplify [latex]\\frac{{x}^{2}-9}{{x}^{2}+4x+3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q568949\">Show Solution<\/button> <\/p>\n<div id=\"q568949\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\begin{array}{lllllllll}\\frac{\\left(x+3\\right)\\left(x - 3\\right)}{\\left(x+3\\right)\\left(x+1\\right)}\\hfill & \\hfill & \\hfill & \\hfill & \\text{Factor the numerator and the denominator}.\\hfill \\\\ \\frac{x - 3}{x+1}\\hfill & \\hfill & \\hfill & \\hfill & \\text{Cancel common factor }\\left(x+3\\right).\\hfill \\end{array}[\/latex]<\/div>\n<p>We can cancel the common factor because any expression divided by itself is equal to [latex]1[\/latex]. <\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox questionHelp\"><strong><strong>Can the [latex]{x}^{2}[\/latex] term be cancelled in the above example?<br \/>\n<\/strong><\/strong><\/p>\n<hr \/>\n<p>No. A factor is an expression that is multiplied by another expression. The [latex]{x}^{2}[\/latex] term is not a factor of the numerator or the denominator.<\/p>\n<\/section>\n<section class=\"textbox example\">Simplify [latex]\\frac{x - 6}{{x}^{2}-36}[\/latex]. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q17752\">Show Solution<\/button> <\/p>\n<div id=\"q17752\" class=\"hidden-answer\" style=\"display: none\"> [latex]\\frac{1}{x+6}[\/latex]<\/div>\n<\/div>\n<\/section>\n<div>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm110917\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110917&theme=lumen&iframe_resize_id=ohm110917&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm110916\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110916&theme=lumen&iframe_resize_id=ohm110916&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_2215_7006\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_2215_7006\"><div tabindex=\"-1\"><p>A trinomial is a type of algebraic expression that consists of exactly three terms. Each term is typically separated by a plus (+) or minus (\u2212) sign.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":12,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex 2:  Identify GCF and Factor a Trinomial\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/3f1RFTIw2Ng\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":280,"module-header":"background_you_need","content_attributions":[{"type":"copyrighted_video","description":"Ex 2:  Identify GCF and Factor a Trinomial","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/3f1RFTIw2Ng","project":"","license":"arr","license_terms":"Standard YouTube License"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=6454717&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hbacgfca-3f1RFTIw2Ng&vembed=0&video_id=3f1RFTIw2Ng&video_target=tpm-plugin-hbacgfca-3f1RFTIw2Ng'><\/script>\n","media_targets":["tpm-plugin-hbacgfca-3f1RFTIw2Ng"]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2215"}],"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":16,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2215\/revisions"}],"predecessor-version":[{"id":7290,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2215\/revisions\/7290"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/280"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2215\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2215"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2215"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2215"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2215"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}