{"id":1460,"date":"2025-07-25T01:56:39","date_gmt":"2025-07-25T01:56:39","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1460"},"modified":"2026-03-24T07:21:16","modified_gmt":"2026-03-24T07:21:16","slug":"partial-fractions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/partial-fractions-fresh-take\/","title":{"raw":"Partial Fractions: Fresh Take","rendered":"Partial Fractions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Decompose P(x)\/Q(x)\u200b\u200b , where Q( x ) has only linear factors.<\/li>\r\n \t<li>Decompose P(x)\/Q(x) , where Q( x ) has irreducible quadratic factors.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Linear Factors<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Concept Overview\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Reverses rational expression addition<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Breaks down complex fraction into simpler parts<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each part has linear denominator<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Must have distinct linear factors<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Degree of numerator &lt; degree of denominator<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General Form For [latex]\\frac{P(x)}{Q(x)}[\/latex] where [latex]Q(x)[\/latex] has distinct linear factors: [latex]\\frac{P(x)}{(a_1x+b_1)(a_2x+b_2)...(a_nx+b_n)} = \\frac{A_1}{a_1x+b_1} + \\frac{A_2}{a_2x+b_2} + ... + \\frac{A_n}{a_nx+b_n}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution Methods\r\nA. Standard Method:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Assign variables to numerators<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiply all terms by common denominator<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Collect like terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve system of equations<\/li>\r\n<\/ul>\r\nB. Heaviside Method:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Substitute values making terms zero<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve for constants one at a time<\/li>\r\n \t<li class=\"whitespace-normal break-words\">More efficient for some problems<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Requirements\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Linear factors must be distinct<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Numerator degree &lt; denominator degree<\/li>\r\n \t<li class=\"whitespace-normal break-words\">All denominators must factor completely<\/li>\r\n \t<li class=\"whitespace-normal break-words\">System must have unique solution<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the partial fraction decomposition of the following expression.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{x}{\\left(x - 3\\right)\\left(x - 2\\right)}[\/latex]<\/p>\r\n[reveal-answer q=\"800291\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"800291\"]\r\n\r\n[latex]\\dfrac{3}{x - 3}-\\dfrac{2}{x - 2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">In this video, you will see another example of how to find a partial fraction decomposition when you have distinct linear factors.\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cdbdeecd-WoVdOcuSI0I\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/WoVdOcuSI0I?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cdbdeecd-WoVdOcuSI0I\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851029&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cdbdeecd-WoVdOcuSI0I&amp;vembed=0&amp;video_id=WoVdOcuSI0I&amp;video_target=tpm-plugin-cdbdeecd-WoVdOcuSI0I\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Partial+Fraction+Decomposition+(Linear+Factors)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Partial Fraction Decomposition (Linear Factors)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Repeated Linear Factors<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Structure for Repeated Factors\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Each repeated factor appears multiple times<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Powers increase from 1 to n (number of repetitions)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General form: [latex]\\frac{P(x)}{Q(x)} = \\frac{A_1}{ax+b} + \\frac{A_2}{(ax+b)^2} + \\frac{A_3}{(ax+b)^3} + ... + \\frac{A_n}{(ax+b)^n}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Requirements\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Must identify repeated factors in denominator<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Numerator degree &lt; denominator degree<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each factor gets unique variable<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Powers must be in increasing order<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution Process\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Factor denominator completely<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Set up partial fractions with all powers<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiply by common denominator<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve system of equations<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verification\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Add all fractions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Should match original expression<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check power requirements<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verify all terms included<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the partial fraction decomposition of the expression with repeated linear factors.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{6x - 11}{{\\left(x - 1\\right)}^{2}}[\/latex]<\/p>\r\n[reveal-answer q=\"623632\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"623632\"]\r\n\r\n[latex]\\dfrac{6}{x - 1}-\\dfrac{5}{{\\left(x - 1\\right)}^{2}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">In this video, you will see an example of how to find the partial fraction decomposition of a rational expression with repeated linear factors.\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-faefbffa-6DdwGw_5dvk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/6DdwGw_5dvk?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-faefbffa-6DdwGw_5dvk\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851030&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-faefbffa-6DdwGw_5dvk&amp;vembed=0&amp;video_id=6DdwGw_5dvk&amp;video_target=tpm-plugin-faefbffa-6DdwGw_5dvk\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+3+-+Partial+Fraction+Decomposition+(Repeated+Linear+Factors)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 3: Partial Fraction Decomposition (Repeated Linear Factors)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Nonrepeated Irreducible Quadratic Factor<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Key Differences from Linear Factors\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Quadratic factor cannot be factored further<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Numerator must be linear ([latex]Ax + B[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each quadratic factor gets unique linear numerator<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Can mix with linear factors<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General Form For [latex]P(x)\/Q(x)[\/latex] with irreducible quadratic factors: [latex]\\frac{P(x)}{Q(x)} = \\frac{A}{ax+b} + \\frac{A_1x + B_1}{a_1x^2 + b_1x + c_1} + \\frac{A_2x + B_2}{a_2x^2 + b_2x + c_2} + ...[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identification Strategy\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Check if quadratic factors are irreducible<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identify any linear factors<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use constants for linear factors<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use linear expressions for quadratic factors<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiple Solution Methods\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Complete system of equations<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Strategic substitution method<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combination of both methods<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Choose most efficient approach<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the partial fraction decomposition of the expression with a nonrepeating irreducible quadratic factor.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{5{x}^{2}-6x+7}{\\left(x - 1\\right)\\left({x}^{2}+1\\right)}[\/latex]<\/p>\r\n[reveal-answer q=\"163296\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"163296\"]\r\n\r\n[latex]\\dfrac{3}{x - 1}+\\dfrac{2x - 4}{{x}^{2}+1}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video, you will see another example of how to find the partial fraction decomposition for a rational expression that has quadratic factors.\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ahcffafd-prtx4o1wbaQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/prtx4o1wbaQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ahcffafd-prtx4o1wbaQ\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851031&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ahcffafd-prtx4o1wbaQ&amp;vembed=0&amp;video_id=prtx4o1wbaQ&amp;video_target=tpm-plugin-ahcffafd-prtx4o1wbaQ\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+5+-+Partial+Fraction+Decomposition+(Linear+and+Quadratic+Factors)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 5: Partial Fraction Decomposition (Linear and Quadratic Factors)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Repeated Irreducible Quadratic Factor<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Structure for Repeated Quadratics\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Each quadratic factor appears multiple times<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Powers increase from [latex]1[\/latex] to [latex]n[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each factor needs linear numerator ([latex]Ax + B[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General form: [latex]\\frac{P(x)}{(ax^2+bx+c)^n} = \\frac{A_1x + B_1}{ax^2+bx+c} + \\frac{A_2x + B_2}{(ax^2+bx+c)^2} + ... + \\frac{A_nx + B_n}{(ax^2+bx+c)^n}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Requirements\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Quadratic must be irreducible<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each power gets unique linear numerator<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Powers must be in increasing order<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Can mix with linear factors<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Numerator Forms\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Linear factors: constant numerator ([latex]A[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Quadratic factors: linear numerator ([latex]Ax + B[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each power needs new variables<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Degree of [latex]P(x) &lt;[\/latex] degree of [latex]Q(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution Process\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Factor denominator completely<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Write decomposition with all powers<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiply by common denominator<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve system of equations<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the partial fraction decomposition of the expression with a repeated irreducible quadratic factor.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{{x}^{3}-4{x}^{2}+9x - 5}{{\\left({x}^{2}-2x+3\\right)}^{2}}[\/latex]<\/p>\r\n[reveal-answer q=\"741991\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"741991\"]\r\n\r\n[latex]\\dfrac{x - 2}{{x}^{2}-2x+3}+\\dfrac{2x+1}{{\\left({x}^{2}-2x+3\\right)}^{2}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">This video provides you with another worked example of how to find the partial fraction decomposition for a rational expression that has repeating quadratic factors.\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cehheaec-Dupeou-FDnI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Dupeou-FDnI?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cehheaec-Dupeou-FDnI\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851032&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cehheaec-Dupeou-FDnI&amp;vembed=0&amp;video_id=Dupeou-FDnI&amp;video_target=tpm-plugin-cehheaec-Dupeou-FDnI\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+6+-+Partial+Fraction+Decomposition+(Repeating+Quadratic+Factors)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 6: Partial Fraction Decomposition (Repeating Quadratic Factors)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Decompose P(x)\/Q(x)\u200b\u200b , where Q( x ) has only linear factors.<\/li>\n<li>Decompose P(x)\/Q(x) , where Q( x ) has irreducible quadratic factors.<\/li>\n<\/ul>\n<\/section>\n<h2>Linear Factors<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Concept Overview\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Reverses rational expression addition<\/li>\n<li class=\"whitespace-normal break-words\">Breaks down complex fraction into simpler parts<\/li>\n<li class=\"whitespace-normal break-words\">Each part has linear denominator<\/li>\n<li class=\"whitespace-normal break-words\">Must have distinct linear factors<\/li>\n<li class=\"whitespace-normal break-words\">Degree of numerator &lt; degree of denominator<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General Form For [latex]\\frac{P(x)}{Q(x)}[\/latex] where [latex]Q(x)[\/latex] has distinct linear factors: [latex]\\frac{P(x)}{(a_1x+b_1)(a_2x+b_2)...(a_nx+b_n)} = \\frac{A_1}{a_1x+b_1} + \\frac{A_2}{a_2x+b_2} + ... + \\frac{A_n}{a_nx+b_n}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Solution Methods<br \/>\nA. Standard Method:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Assign variables to numerators<\/li>\n<li class=\"whitespace-normal break-words\">Multiply all terms by common denominator<\/li>\n<li class=\"whitespace-normal break-words\">Collect like terms<\/li>\n<li class=\"whitespace-normal break-words\">Solve system of equations<\/li>\n<\/ul>\n<p>B. Heaviside Method:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Substitute values making terms zero<\/li>\n<li class=\"whitespace-normal break-words\">Solve for constants one at a time<\/li>\n<li class=\"whitespace-normal break-words\">More efficient for some problems<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Requirements\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Linear factors must be distinct<\/li>\n<li class=\"whitespace-normal break-words\">Numerator degree &lt; denominator degree<\/li>\n<li class=\"whitespace-normal break-words\">All denominators must factor completely<\/li>\n<li class=\"whitespace-normal break-words\">System must have unique solution<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the partial fraction decomposition of the following expression.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{x}{\\left(x - 3\\right)\\left(x - 2\\right)}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q800291\">Show Solution<\/button><\/p>\n<div id=\"q800291\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{3}{x - 3}-\\dfrac{2}{x - 2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In this video, you will see another example of how to find a partial fraction decomposition when you have distinct linear factors.<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cdbdeecd-WoVdOcuSI0I\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/WoVdOcuSI0I?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cdbdeecd-WoVdOcuSI0I\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851029&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cdbdeecd-WoVdOcuSI0I&amp;vembed=0&amp;video_id=WoVdOcuSI0I&amp;video_target=tpm-plugin-cdbdeecd-WoVdOcuSI0I\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Partial+Fraction+Decomposition+(Linear+Factors)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Partial Fraction Decomposition (Linear Factors)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Repeated Linear Factors<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Structure for Repeated Factors\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Each repeated factor appears multiple times<\/li>\n<li class=\"whitespace-normal break-words\">Powers increase from 1 to n (number of repetitions)<\/li>\n<li class=\"whitespace-normal break-words\">General form: [latex]\\frac{P(x)}{Q(x)} = \\frac{A_1}{ax+b} + \\frac{A_2}{(ax+b)^2} + \\frac{A_3}{(ax+b)^3} + ... + \\frac{A_n}{(ax+b)^n}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Requirements\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Must identify repeated factors in denominator<\/li>\n<li class=\"whitespace-normal break-words\">Numerator degree &lt; denominator degree<\/li>\n<li class=\"whitespace-normal break-words\">Each factor gets unique variable<\/li>\n<li class=\"whitespace-normal break-words\">Powers must be in increasing order<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solution Process\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Factor denominator completely<\/li>\n<li class=\"whitespace-normal break-words\">Set up partial fractions with all powers<\/li>\n<li class=\"whitespace-normal break-words\">Multiply by common denominator<\/li>\n<li class=\"whitespace-normal break-words\">Solve system of equations<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Verification\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Add all fractions<\/li>\n<li class=\"whitespace-normal break-words\">Should match original expression<\/li>\n<li class=\"whitespace-normal break-words\">Check power requirements<\/li>\n<li class=\"whitespace-normal break-words\">Verify all terms included<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the partial fraction decomposition of the expression with repeated linear factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{6x - 11}{{\\left(x - 1\\right)}^{2}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q623632\">Show Solution<\/button><\/p>\n<div id=\"q623632\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{6}{x - 1}-\\dfrac{5}{{\\left(x - 1\\right)}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In this video, you will see an example of how to find the partial fraction decomposition of a rational expression with repeated linear factors.<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-faefbffa-6DdwGw_5dvk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/6DdwGw_5dvk?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-faefbffa-6DdwGw_5dvk\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851030&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-faefbffa-6DdwGw_5dvk&amp;vembed=0&amp;video_id=6DdwGw_5dvk&amp;video_target=tpm-plugin-faefbffa-6DdwGw_5dvk\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+3+-+Partial+Fraction+Decomposition+(Repeated+Linear+Factors)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 3: Partial Fraction Decomposition (Repeated Linear Factors)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Nonrepeated Irreducible Quadratic Factor<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Key Differences from Linear Factors\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Quadratic factor cannot be factored further<\/li>\n<li class=\"whitespace-normal break-words\">Numerator must be linear ([latex]Ax + B[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Each quadratic factor gets unique linear numerator<\/li>\n<li class=\"whitespace-normal break-words\">Can mix with linear factors<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General Form For [latex]P(x)\/Q(x)[\/latex] with irreducible quadratic factors: [latex]\\frac{P(x)}{Q(x)} = \\frac{A}{ax+b} + \\frac{A_1x + B_1}{a_1x^2 + b_1x + c_1} + \\frac{A_2x + B_2}{a_2x^2 + b_2x + c_2} + ...[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Identification Strategy\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Check if quadratic factors are irreducible<\/li>\n<li class=\"whitespace-normal break-words\">Identify any linear factors<\/li>\n<li class=\"whitespace-normal break-words\">Use constants for linear factors<\/li>\n<li class=\"whitespace-normal break-words\">Use linear expressions for quadratic factors<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Multiple Solution Methods\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Complete system of equations<\/li>\n<li class=\"whitespace-normal break-words\">Strategic substitution method<\/li>\n<li class=\"whitespace-normal break-words\">Combination of both methods<\/li>\n<li class=\"whitespace-normal break-words\">Choose most efficient approach<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the partial fraction decomposition of the expression with a nonrepeating irreducible quadratic factor.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{5{x}^{2}-6x+7}{\\left(x - 1\\right)\\left({x}^{2}+1\\right)}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q163296\">Show Solution<\/button><\/p>\n<div id=\"q163296\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{3}{x - 1}+\\dfrac{2x - 4}{{x}^{2}+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video, you will see another example of how to find the partial fraction decomposition for a rational expression that has quadratic factors.<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ahcffafd-prtx4o1wbaQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/prtx4o1wbaQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ahcffafd-prtx4o1wbaQ\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851031&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ahcffafd-prtx4o1wbaQ&amp;vembed=0&amp;video_id=prtx4o1wbaQ&amp;video_target=tpm-plugin-ahcffafd-prtx4o1wbaQ\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+5+-+Partial+Fraction+Decomposition+(Linear+and+Quadratic+Factors)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 5: Partial Fraction Decomposition (Linear and Quadratic Factors)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Repeated Irreducible Quadratic Factor<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Structure for Repeated Quadratics\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Each quadratic factor appears multiple times<\/li>\n<li class=\"whitespace-normal break-words\">Powers increase from [latex]1[\/latex] to [latex]n[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Each factor needs linear numerator ([latex]Ax + B[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">General form: [latex]\\frac{P(x)}{(ax^2+bx+c)^n} = \\frac{A_1x + B_1}{ax^2+bx+c} + \\frac{A_2x + B_2}{(ax^2+bx+c)^2} + ... + \\frac{A_nx + B_n}{(ax^2+bx+c)^n}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Requirements\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Quadratic must be irreducible<\/li>\n<li class=\"whitespace-normal break-words\">Each power gets unique linear numerator<\/li>\n<li class=\"whitespace-normal break-words\">Powers must be in increasing order<\/li>\n<li class=\"whitespace-normal break-words\">Can mix with linear factors<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Numerator Forms\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Linear factors: constant numerator ([latex]A[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Quadratic factors: linear numerator ([latex]Ax + B[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Each power needs new variables<\/li>\n<li class=\"whitespace-normal break-words\">Degree of [latex]P(x) <[\/latex] degree of [latex]Q(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solution Process\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Factor denominator completely<\/li>\n<li class=\"whitespace-normal break-words\">Write decomposition with all powers<\/li>\n<li class=\"whitespace-normal break-words\">Multiply by common denominator<\/li>\n<li class=\"whitespace-normal break-words\">Solve system of equations<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the partial fraction decomposition of the expression with a repeated irreducible quadratic factor.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{{x}^{3}-4{x}^{2}+9x - 5}{{\\left({x}^{2}-2x+3\\right)}^{2}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q741991\">Show Solution<\/button><\/p>\n<div id=\"q741991\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{x - 2}{{x}^{2}-2x+3}+\\dfrac{2x+1}{{\\left({x}^{2}-2x+3\\right)}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">This video provides you with another worked example of how to find the partial fraction decomposition for a rational expression that has repeating quadratic factors.<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cehheaec-Dupeou-FDnI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Dupeou-FDnI?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cehheaec-Dupeou-FDnI\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851032&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cehheaec-Dupeou-FDnI&amp;vembed=0&amp;video_id=Dupeou-FDnI&amp;video_target=tpm-plugin-cehheaec-Dupeou-FDnI\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+6+-+Partial+Fraction+Decomposition+(Repeating+Quadratic+Factors)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 6: Partial Fraction Decomposition (Repeating Quadratic Factors)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":32,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex 1: 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