{"id":1225,"date":"2024-05-07T01:44:49","date_gmt":"2024-05-07T01:44:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1225"},"modified":"2024-12-03T18:04:36","modified_gmt":"2024-12-03T18:04:36","slug":"module-4-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/module-4-background-youll-need-3\/","title":{"raw":"Non-Linear Equations: Background You'll Need 3","rendered":"Non-Linear Equations: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Break down polynomial expressions into simpler parts by factoring.<\/span><\/li>\r\n<\/ul>\r\n<\/section>Factoring a polynomial is a method used to break down the polynomial into simpler terms (factors) that, when multiplied together, give back the original polynomial.\r\n\r\nHere\u2019s a general approach to factoring different types of polynomials:\r\n\r\n<strong>1. Factor Out the Greatest Common Factor (GCF)<\/strong>\r\n<ul>\r\n \t<li><strong>Step 1:<\/strong> Identify the greatest common factor among the coefficients and variables in all terms of the polynomial.<\/li>\r\n \t<li><strong>Step 2:<\/strong> Factor out the GCF from each term.<\/li>\r\n<\/ul>\r\n2. <strong>Factoring by Grouping (for polynomials with four or more terms)<\/strong>\r\n<ul>\r\n \t<li><strong>Step 1:<\/strong> Group terms that have common factors.<\/li>\r\n \t<li><strong>Step 2:<\/strong> Factor out the common factor from each group.<\/li>\r\n \t<li><strong>Step 3:<\/strong> If the remaining terms inside the parentheses are the same, factor them out.<\/li>\r\n<\/ul>\r\n3. <strong>Factoring Trinomials<\/strong>\r\n<ul>\r\n \t<li>For trinomials of the form [latex]ax^2+bx+c[\/latex]:\r\n<ul>\r\n \t<li>Step 1: Look for two numbers that multiply to [latex]ac[\/latex] (the product of the coefficient of [latex]x^2[\/latex] and the constant term) and add to [latex]b[\/latex] (the coefficient of [latex]x[\/latex]).<\/li>\r\n \t<li>Step 2: Use these numbers to split the middle term and factor by grouping.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n4. <strong>Factoring Differences of Squares<\/strong>\r\n<ul>\r\n \t<li>For expressions like [latex]a^2 - b^2[\/latex]:\r\n<ul>\r\n \t<li><strong>Step 1:<\/strong> Recognize the pattern [latex]a^2 - b^2 = (a+b)(a-b)[\/latex].<\/li>\r\n \t<li><strong>Step 2:<\/strong> Substitute back the values of [latex]a[\/latex] and [latex]b[\/latex] to factorize.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n5. <strong>Factoring Perfect Square Trinomials<\/strong>\r\n<ul>\r\n \t<li>For trinomials like [latex]a^2+2ab+b^2[\/latex]:\r\n<ul>\r\n \t<li><strong>Step 1:<\/strong> Identify the square roots of the first and last terms.<\/li>\r\n \t<li><strong>Step 2:<\/strong> Ensure the middle term is twice the product of these roots, then factor as [latex](a+b)^2[\/latex] or [latex](a-b)^2[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n6. <strong>Factoring Cubes<\/strong>\r\n<ul>\r\n \t<li>For expressions like [latex]a^3+b^3[\/latex] or [latex]a^3 - b^3[\/latex]:\r\n<ul>\r\n \t<li>Apply the sum or difference of cubes formula: [latex](a^3+b^3) = (a+b)(a^2+ab+b^2)[\/latex] and [latex]a^3-b^3 = (a-b)(a^2+ab+b^2)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<section class=\"textbox example\">Factor the following expression: <center>[latex]6x^2+11x+3[\/latex]<\/center>\r\n\r\n[reveal-answer q=\"255212\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"255212\"] <center>[latex]\\begin{align*} \\text{Given polynomial} &amp; : &amp; 6x^2 + 11x + 3 &amp; &amp; \\\\ \\text{Product \\&amp; Sum} &amp; : &amp; \\text{Find numbers that multiply to } 18 \\text{ and add to } 11. &amp; &amp; \\\\ &amp; &amp; \\text{These numbers are } 9 \\text{ and } 2. &amp; &amp; \\\\ \\text{Rewrite the polynomial} &amp; : &amp; 6x^2 + 9x + 2x + 3 &amp; &amp; \\\\ \\text{Group and factor} &amp; : &amp; (6x^2 + 9x) + (2x + 3) &amp; &amp; \\\\ &amp; &amp; = 3x(2x + 3) + 1(2x + 3) &amp; &amp; \\\\ \\text{Factor out the common term} &amp; : &amp; (3x + 1)(2x + 3) &amp; &amp; \\end{align*}[\/latex]<\/center>[\/hidden-answer]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18999[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]19000[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\">Break down polynomial expressions into simpler parts by factoring.<\/span><\/li>\n<\/ul>\n<\/section>\n<p>Factoring a polynomial is a method used to break down the polynomial into simpler terms (factors) that, when multiplied together, give back the original polynomial.<\/p>\n<p>Here\u2019s a general approach to factoring different types of polynomials:<\/p>\n<p><strong>1. Factor Out the Greatest Common Factor (GCF)<\/strong><\/p>\n<ul>\n<li><strong>Step 1:<\/strong> Identify the greatest common factor among the coefficients and variables in all terms of the polynomial.<\/li>\n<li><strong>Step 2:<\/strong> Factor out the GCF from each term.<\/li>\n<\/ul>\n<p>2. <strong>Factoring by Grouping (for polynomials with four or more terms)<\/strong><\/p>\n<ul>\n<li><strong>Step 1:<\/strong> Group terms that have common factors.<\/li>\n<li><strong>Step 2:<\/strong> Factor out the common factor from each group.<\/li>\n<li><strong>Step 3:<\/strong> If the remaining terms inside the parentheses are the same, factor them out.<\/li>\n<\/ul>\n<p>3. <strong>Factoring Trinomials<\/strong><\/p>\n<ul>\n<li>For trinomials of the form [latex]ax^2+bx+c[\/latex]:\n<ul>\n<li>Step 1: Look for two numbers that multiply to [latex]ac[\/latex] (the product of the coefficient of [latex]x^2[\/latex] and the constant term) and add to [latex]b[\/latex] (the coefficient of [latex]x[\/latex]).<\/li>\n<li>Step 2: Use these numbers to split the middle term and factor by grouping.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>4. <strong>Factoring Differences of Squares<\/strong><\/p>\n<ul>\n<li>For expressions like [latex]a^2 - b^2[\/latex]:\n<ul>\n<li><strong>Step 1:<\/strong> Recognize the pattern [latex]a^2 - b^2 = (a+b)(a-b)[\/latex].<\/li>\n<li><strong>Step 2:<\/strong> Substitute back the values of [latex]a[\/latex] and [latex]b[\/latex] to factorize.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>5. <strong>Factoring Perfect Square Trinomials<\/strong><\/p>\n<ul>\n<li>For trinomials like [latex]a^2+2ab+b^2[\/latex]:\n<ul>\n<li><strong>Step 1:<\/strong> Identify the square roots of the first and last terms.<\/li>\n<li><strong>Step 2:<\/strong> Ensure the middle term is twice the product of these roots, then factor as [latex](a+b)^2[\/latex] or [latex](a-b)^2[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>6. <strong>Factoring Cubes<\/strong><\/p>\n<ul>\n<li>For expressions like [latex]a^3+b^3[\/latex] or [latex]a^3 - b^3[\/latex]:\n<ul>\n<li>Apply the sum or difference of cubes formula: [latex](a^3+b^3) = (a+b)(a^2+ab+b^2)[\/latex] and [latex]a^3-b^3 = (a-b)(a^2+ab+b^2)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<section class=\"textbox example\">Factor the following expression: <\/p>\n<div style=\"text-align: center;\">[latex]6x^2+11x+3[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q255212\">Show Answer<\/button><\/p>\n<div id=\"q255212\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\begin{align*} \\text{Given polynomial} & : & 6x^2 + 11x + 3 & & \\\\ \\text{Product \\& Sum} & : & \\text{Find numbers that multiply to } 18 \\text{ and add to } 11. & & \\\\ & & \\text{These numbers are } 9 \\text{ and } 2. & & \\\\ \\text{Rewrite the polynomial} & : & 6x^2 + 9x + 2x + 3 & & \\\\ \\text{Group and factor} & : & (6x^2 + 9x) + (2x + 3) & & \\\\ & & = 3x(2x + 3) + 1(2x + 3) & & \\\\ \\text{Factor out the common term} & : & (3x + 1)(2x + 3) & & \\end{align*}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18999\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18999&theme=lumen&iframe_resize_id=ohm18999&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm19000\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=19000&theme=lumen&iframe_resize_id=ohm19000&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":92,"module-header":"background_you_need","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\/1225"}],"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":9,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1225\/revisions"}],"predecessor-version":[{"id":6585,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1225\/revisions\/6585"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/92"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1225\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=1225"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1225"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=1225"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=1225"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}