{"id":936,"date":"2024-04-30T21:42:22","date_gmt":"2024-04-30T21:42:22","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=936"},"modified":"2025-02-01T06:40:03","modified_gmt":"2025-02-01T06:40:03","slug":"rational-expressions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/rational-expressions-fresh-take\/","title":{"raw":"Rational Expressions: Fresh Take","rendered":"Rational Expressions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Simplify, multiply, and divide rational expressions.<\/li>\r\n \t<li>Add and subtract rational expressions, making sure to correctly handle the denominators.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Rational Expressions<\/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\">Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A rational expression is a fraction of polynomials: [latex]\\frac{P(x)}{Q(x)}[\/latex] where [latex]P(x)[\/latex] and [latex]Q(x)[\/latex] are polynomials<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplification Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Factor both numerator and denominator<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Cancel common factors<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Concept:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Only cancel factors, not individual terms<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Simplify [latex]\\dfrac{x - 6}{{x}^{2}-36}[\/latex].[reveal-answer q=\"17752\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"17752\"][latex]\\frac{1}{x+6}[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-acdeebgc-uVpsz-xpnPo\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/uVpsz-xpnPo?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-acdeebgc-uVpsz-xpnPo\" 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=13013589&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-acdeebgc-uVpsz-xpnPo&vembed=0&video_id=uVpsz-xpnPo&video_target=tpm-plugin-acdeebgc-uVpsz-xpnPo'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Simplifying+Rational+Expressions_transcript.txt\" target=\"_blank\" rel=\"noopener\"> transcript for \u201cSimplifying Rational Expressions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Multiplying Rational Expressions<\/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>Key Concept:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Multiplication of rational expressions follows the same rules as multiplication of fractions<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">\u00a0Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Factor numerators and denominators<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiply numerators together<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiply denominators together<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify the result<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplification:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Cancel common factors between numerator and denominator before multiplying<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Multiply the rational expressions and show the product in simplest form:\r\n<div style=\"text-align: center;\">[latex]\\dfrac{{x}^{2}+11x+30}{{x}^{2}+5x+6}\\cdot \\dfrac{{x}^{2}+7x+12}{{x}^{2}+8x+16}[\/latex]<\/div>\r\n[reveal-answer q=\"165135\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"165135\"]\r\n\r\n[latex]\\dfrac{\\left(x+5\\right)\\left(x+6\\right)}{\\left(x+2\\right)\\left(x+4\\right)}[\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ffcdhfhe-pu4dkSEoWZY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/pu4dkSEoWZY?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ffcdhfhe-pu4dkSEoWZY\" 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=12779052&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ffcdhfhe-pu4dkSEoWZY&vembed=0&video_id=pu4dkSEoWZY&video_target=tpm-plugin-ffcdhfhe-pu4dkSEoWZY'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Multiply+Rational+Expressions+with+Restrictions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMultiply Rational Expressions with Restrictions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Dividing Rational Expressions<\/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 Concept:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Division of rational expressions is equivalent to multiplication by the reciprocal<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Rewrite as multiplication by reciprocal<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor numerators and denominators<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiply numerators together<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiply denominators together<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify the result<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Formula<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Divide the rational expressions and express the quotient in simplest form:\r\n<div style=\"text-align: center;\">[latex]\\dfrac{9{x}^{2}-16}{3{x}^{2}+17x - 28}\\div \\dfrac{3{x}^{2}-2x - 8}{{x}^{2}+5x - 14}[\/latex]<\/div>\r\n[reveal-answer q=\"396693\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"396693\"]\r\n\r\n[latex]1[\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-eedhdhca-0cmhBgQNGDI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/0cmhBgQNGDI?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-eedhdhca-0cmhBgQNGDI\" 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=12844186&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-eedhdhca-0cmhBgQNGDI&vembed=0&video_id=0cmhBgQNGDI&video_target=tpm-plugin-eedhdhca-0cmhBgQNGDI'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Dividing+rational+expressions+%7C+Precalculus+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDividing rational expressions | Precalculus | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Adding and Subtracting Rational Expressions<\/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>Key Concept:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Addition and subtraction of rational expressions follow the same rules as addition and subtraction of fractions<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">\u00a0Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Find the Least Common Denominator (LCD)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rewrite expressions with the LCD<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Add or subtract the numerators<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify the result<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Least Common Denominator (LCD):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Smallest multiple that the denominators have in common<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Found by factoring denominators and multiplying all distinct factors<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Add the rational expressions: [latex]\\dfrac{2}{x-1} + \\dfrac{3}{x+2}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[reveal-answer q=\"604173\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"604173\"]<\/p>\r\n\r\n<ol>\r\n \t<li class=\"whitespace-normal break-words\">Find the LCD: <a href=\"x-1\">latex<\/a>(x+2)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rewrite expressions with LCD: [latex]\\frac{2}{x-1} \\cdot \\frac{x+2}{x+2} + \\frac{3}{x+2} \\cdot \\frac{x-1}{x-1}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiply numerators and denominators: [latex]\\frac{2(x+2)}{(x-1)(x+2)} + \\frac{3(x-1)}{(x-1)(x+2)}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Add numerators: [latex]\\frac{2(x+2) + 3(x-1)}{(x-1)(x+2)}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify: [latex]\\frac{2x+4 + 3x-3}{(x-1)(x+2)} = \\frac{5x+1}{(x-1)(x+2)}[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gahgdgeb-DrE_iiw1Mvk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/DrE_iiw1Mvk?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gahgdgeb-DrE_iiw1Mvk\" 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=12844187&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gahgdgeb-DrE_iiw1Mvk&vembed=0&video_id=DrE_iiw1Mvk&video_target=tpm-plugin-gahgdgeb-DrE_iiw1Mvk'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Adding+rational+expression+-+unlike+denominators+%7C+High+School+Math+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAdding rational expression: unlike denominators | High School Math | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Subtract the rational expressions: [latex]\\dfrac{3}{x+5}-\\dfrac{1}{x - 3}[\/latex].[reveal-answer q=\"820348\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"820348\"][latex]\\dfrac{2\\left(x - 7\\right)}{\\left(x+5\\right)\\left(x - 3\\right)}[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hcagdbff-evmDZkDvlNw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/evmDZkDvlNw?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hcagdbff-evmDZkDvlNw\" 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=12844188&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hcagdbff-evmDZkDvlNw&vembed=0&video_id=evmDZkDvlNw&video_target=tpm-plugin-hcagdbff-evmDZkDvlNw'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Subtracting+rational+expressions+-+unlike+denominators+%7C+High+School+Math+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSubtracting rational expressions: unlike denominators | High School Math | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Simplifying Complex Rational Expressions<\/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>Key Concept:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Every complex rational expression can be simplified to a standard rational expression<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>\u00a0Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A complex rational expression is a fraction that contains one or more fractions in its numerator, denominator, or both<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplification Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Combine expressions in the numerator into a single fraction<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combine expressions in the denominator into a single fraction<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Divide the numerator by the denominator<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rewrite as multiplication by the reciprocal<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiply and simplify<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Technique:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Use the LCD method to combine fractions within the numerator or denominator<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Simplify: [latex]\\dfrac{\\dfrac{x}{y}-\\dfrac{y}{x}}{y}[\/latex][reveal-answer q=\"40643\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"40643\"][latex]\\dfrac{{x}^{2}-{y}^{2}}{x{y}^{2}}[\/latex][\/hidden-answer]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Simplify, multiply, and divide rational expressions.<\/li>\n<li>Add and subtract rational expressions, making sure to correctly handle the denominators.<\/li>\n<\/ul>\n<\/section>\n<h2>Rational Expressions<\/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\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A rational expression is a fraction of polynomials: [latex]\\frac{P(x)}{Q(x)}[\/latex] where [latex]P(x)[\/latex] and [latex]Q(x)[\/latex] are polynomials<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Simplification Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Factor both numerator and denominator<\/li>\n<li class=\"whitespace-normal break-words\">Cancel common factors<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Concept:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Only cancel factors, not individual terms<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Simplify [latex]\\dfrac{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<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-acdeebgc-uVpsz-xpnPo\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/uVpsz-xpnPo?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-acdeebgc-uVpsz-xpnPo\" 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=13013589&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-acdeebgc-uVpsz-xpnPo&#38;vembed=0&#38;video_id=uVpsz-xpnPo&#38;video_target=tpm-plugin-acdeebgc-uVpsz-xpnPo\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Simplifying+Rational+Expressions_transcript.txt\" target=\"_blank\" rel=\"noopener\"> transcript for \u201cSimplifying Rational Expressions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Multiplying Rational Expressions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li>Key Concept:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Multiplication of rational expressions follows the same rules as multiplication of fractions<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">\u00a0Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Factor numerators and denominators<\/li>\n<li class=\"whitespace-normal break-words\">Multiply numerators together<\/li>\n<li class=\"whitespace-normal break-words\">Multiply denominators together<\/li>\n<li class=\"whitespace-normal break-words\">Simplify the result<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Simplification:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Cancel common factors between numerator and denominator before multiplying<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Multiply the rational expressions and show the product in simplest form:<\/p>\n<div style=\"text-align: center;\">[latex]\\dfrac{{x}^{2}+11x+30}{{x}^{2}+5x+6}\\cdot \\dfrac{{x}^{2}+7x+12}{{x}^{2}+8x+16}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q165135\">Show Solution<\/button><\/p>\n<div id=\"q165135\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{\\left(x+5\\right)\\left(x+6\\right)}{\\left(x+2\\right)\\left(x+4\\right)}[\/latex]<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ffcdhfhe-pu4dkSEoWZY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/pu4dkSEoWZY?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ffcdhfhe-pu4dkSEoWZY\" 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=12779052&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ffcdhfhe-pu4dkSEoWZY&#38;vembed=0&#38;video_id=pu4dkSEoWZY&#38;video_target=tpm-plugin-ffcdhfhe-pu4dkSEoWZY\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Multiply+Rational+Expressions+with+Restrictions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMultiply Rational Expressions with Restrictions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Dividing Rational Expressions<\/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 Concept:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Division of rational expressions is equivalent to multiplication by the reciprocal<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Rewrite as multiplication by reciprocal<\/li>\n<li class=\"whitespace-normal break-words\">Factor numerators and denominators<\/li>\n<li class=\"whitespace-normal break-words\">Multiply numerators together<\/li>\n<li class=\"whitespace-normal break-words\">Multiply denominators together<\/li>\n<li class=\"whitespace-normal break-words\">Simplify the result<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Formula<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Divide the rational expressions and express the quotient in simplest form:<\/p>\n<div style=\"text-align: center;\">[latex]\\dfrac{9{x}^{2}-16}{3{x}^{2}+17x - 28}\\div \\dfrac{3{x}^{2}-2x - 8}{{x}^{2}+5x - 14}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q396693\">Show Solution<\/button><\/p>\n<div id=\"q396693\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]1[\/latex]<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-eedhdhca-0cmhBgQNGDI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/0cmhBgQNGDI?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-eedhdhca-0cmhBgQNGDI\" 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=12844186&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-eedhdhca-0cmhBgQNGDI&#38;vembed=0&#38;video_id=0cmhBgQNGDI&#38;video_target=tpm-plugin-eedhdhca-0cmhBgQNGDI\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Dividing+rational+expressions+%7C+Precalculus+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDividing rational expressions | Precalculus | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Adding and Subtracting Rational Expressions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li>Key Concept:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Addition and subtraction of rational expressions follow the same rules as addition and subtraction of fractions<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">\u00a0Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Find the Least Common Denominator (LCD)<\/li>\n<li class=\"whitespace-normal break-words\">Rewrite expressions with the LCD<\/li>\n<li class=\"whitespace-normal break-words\">Add or subtract the numerators<\/li>\n<li class=\"whitespace-normal break-words\">Simplify the result<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Least Common Denominator (LCD):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Smallest multiple that the denominators have in common<\/li>\n<li class=\"whitespace-normal break-words\">Found by factoring denominators and multiplying all distinct factors<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Add the rational expressions: [latex]\\dfrac{2}{x-1} + \\dfrac{3}{x+2}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q604173\">Show Answer<\/button><\/p>\n<div id=\"q604173\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li class=\"whitespace-normal break-words\">Find the LCD: <a href=\"x-1\">latex<\/a>(x+2)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Rewrite expressions with LCD: [latex]\\frac{2}{x-1} \\cdot \\frac{x+2}{x+2} + \\frac{3}{x+2} \\cdot \\frac{x-1}{x-1}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Multiply numerators and denominators: [latex]\\frac{2(x+2)}{(x-1)(x+2)} + \\frac{3(x-1)}{(x-1)(x+2)}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Add numerators: [latex]\\frac{2(x+2) + 3(x-1)}{(x-1)(x+2)}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Simplify: [latex]\\frac{2x+4 + 3x-3}{(x-1)(x+2)} = \\frac{5x+1}{(x-1)(x+2)}[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gahgdgeb-DrE_iiw1Mvk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/DrE_iiw1Mvk?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gahgdgeb-DrE_iiw1Mvk\" 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=12844187&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-gahgdgeb-DrE_iiw1Mvk&#38;vembed=0&#38;video_id=DrE_iiw1Mvk&#38;video_target=tpm-plugin-gahgdgeb-DrE_iiw1Mvk\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Adding+rational+expression+-+unlike+denominators+%7C+High+School+Math+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAdding rational expression: unlike denominators | High School Math | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Subtract the rational expressions: [latex]\\dfrac{3}{x+5}-\\dfrac{1}{x - 3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q820348\">Show Solution<\/button><\/p>\n<div id=\"q820348\" class=\"hidden-answer\" style=\"display: none\">[latex]\\dfrac{2\\left(x - 7\\right)}{\\left(x+5\\right)\\left(x - 3\\right)}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hcagdbff-evmDZkDvlNw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/evmDZkDvlNw?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hcagdbff-evmDZkDvlNw\" 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=12844188&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-hcagdbff-evmDZkDvlNw&#38;vembed=0&#38;video_id=evmDZkDvlNw&#38;video_target=tpm-plugin-hcagdbff-evmDZkDvlNw\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Subtracting+rational+expressions+-+unlike+denominators+%7C+High+School+Math+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSubtracting rational expressions: unlike denominators | High School Math | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Simplifying Complex Rational Expressions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li>Key Concept:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Every complex rational expression can be simplified to a standard rational expression<\/li>\n<\/ul>\n<\/li>\n<li>\u00a0Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A complex rational expression is a fraction that contains one or more fractions in its numerator, denominator, or both<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Simplification Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Combine expressions in the numerator into a single fraction<\/li>\n<li class=\"whitespace-normal break-words\">Combine expressions in the denominator into a single fraction<\/li>\n<li class=\"whitespace-normal break-words\">Divide the numerator by the denominator<\/li>\n<li class=\"whitespace-normal break-words\">Rewrite as multiplication by the reciprocal<\/li>\n<li class=\"whitespace-normal break-words\">Multiply and simplify<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Technique:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use the LCD method to combine fractions within the numerator or denominator<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Simplify: [latex]\\dfrac{\\dfrac{x}{y}-\\dfrac{y}{x}}{y}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q40643\">Show Solution<\/button><\/p>\n<div id=\"q40643\" class=\"hidden-answer\" style=\"display: none\">[latex]\\dfrac{{x}^{2}-{y}^{2}}{x{y}^{2}}[\/latex]<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":12,"menu_order":22,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Simplifying Rational Expressions\",\"author\":\"\",\"organization\":\"The Organic Chemistry Tutor\",\"url\":\"https:\/\/youtu.be\/uVpsz-xpnPo\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Multiply Rational Expressions with Restrictions \",\"author\":\"\",\"organization\":\"Anil Kumar\",\"url\":\"https:\/\/www.youtube.com\/watch?v=pu4dkSEoWZY\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Dividing rational expressions | Precalculus | Khan Academy \",\"author\":\"\",\"organization\":\"Khan 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