{"id":2076,"date":"2025-09-03T14:33:06","date_gmt":"2025-09-03T14:33:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=2076"},"modified":"2025-09-03T14:51:20","modified_gmt":"2025-09-03T14:51:20","slug":"advanced-integration-techniques-background-youll-need-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/advanced-integration-techniques-background-youll-need-4\/","title":{"raw":"Advanced Integration Techniques: Background You\u2019ll Need 4","rendered":"Advanced Integration Techniques: Background You\u2019ll Need 4"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Factor polynomial expressions into their simplest linear and quadratic parts<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Factoring Polynomial Expressions<\/h2>\r\nFactoring a polynomial is a method used to break down the polynomial into simpler terms (factors) that, when multiplied together, give back the original polynomial. <strong>Complete factorization<\/strong> means continuing until all factors are either linear or irreducible quadratic factors.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>complete factorization<\/h3>\r\nA polynomial is completely factored when it's written as a product of linear factors (like [latex](x - 3)[\/latex]) and irreducible quadratic factors (quadratics that cannot be factored further over the real numbers).\r\n\r\n<\/section>Not all quadratic expressions can be factored into linear factors with real numbers. A quadratic [latex]ax^2 + bx + c[\/latex] is <strong>irreducible<\/strong> when its discriminant [latex]\\Delta = b^2 - 4ac &lt; 0[\/latex].\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Testing for Irreducibility<\/strong>: Calculate [latex]b^2 - 4ac[\/latex] for any quadratic factor. If this value is negative, the quadratic cannot be factored further over the real numbers and you've reached complete factorization for that factor.<\/section>\r\n<p class=\"whitespace-normal break-words\">Follow these steps to achieve complete factorization:<\/p>\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 (for trinomials of the form [latex]ax^2+bx+c[\/latex)<\/strong>\r\n<ul>\r\n \t<li><strong>Step 1:<\/strong> 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><strong>Step 2<\/strong>: Use these numbers to split the middle term and factor by grouping.<\/li>\r\n \t<li><strong>Step 3:<\/strong> Check if the result can be factored further using the discriminant test.<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">4.<strong> Factoring Special Patterns<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Difference of Squares:<\/strong> [latex]a^2 - b^2 = (a+b)(a-b)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Perfect Square Trinomials:<\/strong> [latex]a^2 \\pm 2ab + b^2 = (a \\pm b)^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Sum\/Difference of Cubes:<\/strong> [latex]a^3 \\pm b^3 = (a \\pm b)(a^2 \\mp ab + b^2)[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">5. <strong>Verify Complete Factorization<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Check that all quadratic factors are irreducible using the discriminant test<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Ensure no further factoring is possible<\/li>\r\n<\/ul>\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p class=\"whitespace-normal break-words\">You've completely factored a polynomial when every factor is either:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">A linear factor (degree 1)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">An irreducible quadratic factor (degree 2 with negative discriminant)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">A constant factor<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\">Factor the following expression:<center>[latex]6x^2+11x+3[\/latex]<\/center>[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\" aria-label=\"Learning Goals\">\n<ul>\n<li>Factor polynomial expressions into their simplest linear and quadratic parts<\/li>\n<\/ul>\n<\/section>\n<h2 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Factoring Polynomial Expressions<\/h2>\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. <strong>Complete factorization<\/strong> means continuing until all factors are either linear or irreducible quadratic factors.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>complete factorization<\/h3>\n<p>A polynomial is completely factored when it&#8217;s written as a product of linear factors (like [latex](x - 3)[\/latex]) and irreducible quadratic factors (quadratics that cannot be factored further over the real numbers).<\/p>\n<\/section>\n<p>Not all quadratic expressions can be factored into linear factors with real numbers. A quadratic [latex]ax^2 + bx + c[\/latex] is <strong>irreducible<\/strong> when its discriminant [latex]\\Delta = b^2 - 4ac < 0[\/latex].\n\n\n\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Testing for Irreducibility<\/strong>: Calculate [latex]b^2 - 4ac[\/latex] for any quadratic factor. If this value is negative, the quadratic cannot be factored further over the real numbers and you&#8217;ve reached complete factorization for that factor.<\/section>\n<p class=\"whitespace-normal break-words\">Follow these steps to achieve complete factorization:<\/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 (for trinomials of the form [latex]ax^2+bx+c[\/latex)<\/strong>  <\/p>\n<ul>\n<li><strong>Step 1:<\/strong> 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><strong>Step 2<\/strong>: Use these numbers to split the middle term and factor by grouping.<\/li>\n<li><strong>Step 3:<\/strong> Check if the result can be factored further using the discriminant test.<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">4.<strong> Factoring Special Patterns<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Difference of Squares:<\/strong> [latex]a^2 - b^2 = (a+b)(a-b)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Perfect Square Trinomials:<\/strong> [latex]a^2 \\pm 2ab + b^2 = (a \\pm b)^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Sum\/Difference of Cubes:<\/strong> [latex]a^3 \\pm b^3 = (a \\pm b)(a^2 \\mp ab + b^2)[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">5. <strong>Verify Complete Factorization<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Check that all quadratic factors are irreducible using the discriminant test<\/li>\n<li class=\"whitespace-normal break-words\">Ensure no further factoring is possible<\/li>\n<\/ul>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p class=\"whitespace-normal break-words\">You've completely factored a polynomial when every factor is either:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">A linear factor (degree 1)<\/li>\n<li class=\"whitespace-normal break-words\">An irreducible quadratic factor (degree 2 with negative discriminant)<\/li>\n<li class=\"whitespace-normal break-words\">A constant factor<\/li>\n<\/ul>\n<\/section>\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":15,"menu_order":5,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":541,"module-header":"- Select Header -","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\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/2076"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/2076\/revisions"}],"predecessor-version":[{"id":2084,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/2076\/revisions\/2084"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/541"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/2076\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=2076"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=2076"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=2076"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=2076"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}