{"id":1353,"date":"2024-05-10T02:50:05","date_gmt":"2024-05-10T02:50:05","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1353"},"modified":"2025-01-16T21:38:52","modified_gmt":"2025-01-16T21:38:52","slug":"other-types-of-equations-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/other-types-of-equations-fresh-take\/","title":{"raw":"Other Types of Equations: Fresh Take","rendered":"Other Types of Equations: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Solve equations that include fractions with variables.<\/li>\r\n \t<li>Solve equations with roots and fractional powers.<\/li>\r\n \t<li>Use factoring to find solutions to polynomial equations.<\/li>\r\n \t<li>Find solutions to equations that involve absolute values.<\/li>\r\n \t<li><span data-sheets-root=\"1\">Find solutions to inequalities that involve absolute values.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Solving a Rational Equation<\/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 of Rational Equations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Contains at least one rational expression<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Variable appears in at least one denominator<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solving Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Factor all denominators<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find and exclude values that make denominators zero<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determine the Least Common Denominator (LCD)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiply both sides by the LCD<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the resulting equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check solutions in the original equation<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Cross-Multiplication Method:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For equations in the form [latex]\\frac{a}{b} = \\frac{c}{d}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiply to get [latex]ad = bc[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Handling Binomial Denominators:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Treat binomials (e.g., [latex]x + 1[\/latex]) as single units<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor completely before finding LCD<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Importance in Algebra:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Bridge between linear equations and more complex algebraic structures<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Foundation for solving many real-world problems<strong>\u00a0<\/strong><\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve the rational equation:\r\n<p style=\"text-align: center;\">[latex]\\dfrac{2}{3x} = \\dfrac{1}{4} - \\dfrac{1}{6x}[\/latex]<\/p>\r\n[reveal-answer q=\"407050\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"407050\"][latex]x=\\dfrac{10}{3}[\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve the rational equation:\r\n<p style=\"text-align: center;\">[latex]-\\dfrac{5}{2x} + \\dfrac{3}{4x} = -\\dfrac{7}{4}[\/latex]<\/p>\r\n[reveal-answer q=\"18903\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"18903\"][latex]x=1[\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve [latex]\\dfrac{-3}{2x+1} = \\dfrac{4}{3x+1}[\/latex]. State the excluded values.[reveal-answer q=\"132629\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"132629\"][latex]x=-\\dfrac{7}{17}[\/latex]. Excluded values are [latex]x= -\\dfrac{1}{2}[\/latex] and [latex]x= -\\dfrac{1}{3}[\/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-bbahfceh-UBfIQnTYGoY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/UBfIQnTYGoY?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bbahfceh-UBfIQnTYGoY\" 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=12844260&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bbahfceh-UBfIQnTYGoY&vembed=0&video_id=UBfIQnTYGoY&video_target=tpm-plugin-bbahfceh-UBfIQnTYGoY'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+2+-+Solving+Rational+Equations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Solving Rational Equations\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Radical Equations<\/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\">Equations containing variables under a radical symbol<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Example: [latex]\\sqrt{3x + 18} = x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solving Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Isolate the radical term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Raise both sides to the power of the radical's index<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the resulting equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check for extraneous solutions<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Extraneous Solutions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Solutions that satisfy the altered equation but not the original<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Result from squaring or cubing both sides<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Must be checked by substitution in the original equation<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiple Radicals:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Isolate one radical at a time<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Repeat the process for each radical<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Strategy<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify all radical terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plan the isolation sequence for multiple radicals<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Raise both sides to appropriate powers<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the resulting polynomial equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check all solutions in the original equation<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve the radical equation: [latex]\\sqrt{x+3}=3x - 1[\/latex][reveal-answer q=\"719648\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"719648\"][latex]x=1[\/latex]; extraneous solution [latex]x=-\\frac{2}{9}[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve the equation with two radicals: [latex]\\sqrt{3x+7}+\\sqrt{x+2}=1[\/latex].[reveal-answer q=\"265496\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"265496\"][latex]x=-2[\/latex]; extraneous solution [latex]x=-1[\/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-dfeaehce-RK0lFf43gSY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/RK0lFf43gSY?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dfeaehce-RK0lFf43gSY\" 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=12844261&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dfeaehce-RK0lFf43gSY&vembed=0&video_id=RK0lFf43gSY&video_target=tpm-plugin-dfeaehce-RK0lFf43gSY'><\/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+-+Solve+Radical+Equations+-+Square+Roots_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 3: Solve Radical Equations - Square Roots\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Solve Equations With Rational Exponents<\/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\">Rational Exponents:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Fractions as exponents<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Notation: [latex]a^{\\frac{m}{n}} = \\sqrt[n]{a^m}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Equivalence to Radicals:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]a^{\\frac{1}{n}} = \\sqrt[n]{a}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a^{\\frac{m}{n}} = (\\sqrt[n]{a})^m[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solving Strategy:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Raise both sides to the reciprocal power<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify using exponent rules<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the resulting equation<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Exponent Rules:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Product: [latex]a^m \\cdot a^n = a^{m+n}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Quotient: [latex]\\frac{a^m}{a^n} = a^{m-n}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Power: [latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Technique<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the rational exponent<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determine its reciprocal<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply the reciprocal exponent to both sides<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify using exponent rules<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the resulting equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check the solution<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate [latex]{64}^{-\\frac{1}{3}}[\/latex].[reveal-answer q=\"68783\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"68783\"][latex]\\frac{1}{4}[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve the equation [latex]{x}^{\\frac{3}{2}}=125[\/latex].[reveal-answer q=\"390459\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"390459\"][latex]25[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve: [latex]{\\left(x+5\\right)}^{\\frac{3}{2}}=8[\/latex].[reveal-answer q=\"943422\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"943422\"][latex]-1[\/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-hhdhceha-tS6J8Vb81RE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/tS6J8Vb81RE?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hhdhceha-tS6J8Vb81RE\" 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=12844262&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hhdhceha-tS6J8Vb81RE&vembed=0&video_id=tS6J8Vb81RE&video_target=tpm-plugin-hhdhceha-tS6J8Vb81RE'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Solve+Equations+with+Rational+Exponents+(Two+Solutions)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolve Equations with Rational Exponents (Two Solutions)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Polynomial Equations<\/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: A polynomial equation is an equation of the form: [latex]a_nx^n + a_{n-1}x^{n-1} + \\ldots + a_2x^2 + a_1x + a_0 = 0[\/latex] where [latex]n[\/latex] is a positive integer and [latex]a_n \\neq 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Degree:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The highest power of the variable in the polynomial<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determines the maximum number of solutions<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Zero-Product Property: If [latex]ab = 0[\/latex], then [latex]a = 0[\/latex] or [latex]b = 0[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Fundamental to solving polynomial equations by factoring<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution Types:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Real solutions (rational or irrational)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Complex solutions (when real solutions don't exist)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Steps<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Arrange the polynomial in standard form (descending powers)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor out the greatest common factor (GCF)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Look for special patterns or grouping opportunities<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor completely<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply the zero-product property<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the resulting linear equations<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check solutions in the original equation<\/li>\r\n<\/ol>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve:\r\n<p style=\"text-align: center;\">[latex]2x^4 - 18x^2 + 40 = 0[\/latex]<\/p>\r\n[reveal-answer q=\"262759\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"262759\"]\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Factor out GCF: [latex]2(x^4 - 9x^2 + 20) = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Recognize [latex]x^4 - 9x^2 + 20[\/latex] as a quadratic in [latex]x^2[\/latex] Let [latex]u = x^2[\/latex], then [latex]u^2 - 9u + 20 = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor the quadratic in [latex]u[\/latex]: [latex]2(u - 4)(u - 5) = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute back [latex]x^2[\/latex] for [latex]u[\/latex]: [latex]2(x^2 - 4)(x^2 - 5) = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply zero-product property: [latex]2 = 0[\/latex] (always false) [latex]x^2 - 4 = 0[\/latex] or [latex]x^2 - 5 = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve resulting equations: [latex]x = \\pm 2[\/latex] or [latex]x = \\pm \\sqrt{5}[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the solutions are [latex]x = 2, -2, \\sqrt{5}, -\\sqrt{5}[\/latex]<\/p>\r\n[\/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-debhabcg-5tiQN3wQZfs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/5tiQN3wQZfs?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-debhabcg-5tiQN3wQZfs\" 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=12844263&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-debhabcg-5tiQN3wQZfs&vembed=0&video_id=5tiQN3wQZfs&video_target=tpm-plugin-debhabcg-5tiQN3wQZfs'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Factor+and+Solve+a+Polynomial+Equation_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Factor and Solve a Polynomial Equation\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Absolute Value Equations<\/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 of Absolute Value:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Represents the distance of a number from zero on the number line<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Always non-negative<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Formally defined as: [latex]|x| = \\begin{cases} x &amp; \\text{if } x \\geq 0 \\\\ -x &amp; \\text{if } x &lt; 0 \\end{cases}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Absolute Value Equation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">An equation containing an absolute value expression<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General form: [latex]|A| = B[\/latex], where A is an expression and B is a non-negative number<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Properties of Absolute Value Equations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">If [latex]|A| = B[\/latex], then [latex]A = B[\/latex] or [latex]A = -B[\/latex] when [latex]B \\geq 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]B &lt; 0[\/latex], the equation [latex]|A| = B[\/latex] has no solution<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Standard Form of Linear Absolute Value Equations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]|ax + b| = c[\/latex], where [latex]a \\neq 0[\/latex] and [latex]c[\/latex] is a real number<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Number of Solutions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">If [latex]c &lt; 0[\/latex]: No solution<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]c = 0[\/latex]: One solution<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]c &gt; 0[\/latex]: Two solutions<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>Solving Process:<\/strong>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Isolate the absolute value expression on one side of the equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Consider two cases: positive and negative<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve each case as a linear equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check solutions in the original equation<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve the absolute value equation: [latex]|1 - 4x|+8=13[\/latex].[reveal-answer q=\"567620\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"567620\"][latex]x=-1[\/latex], [latex]x=\\frac{3}{2}[\/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-gghgchfg-sBR_ontVFXU\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/sBR_ontVFXU?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gghgchfg-sBR_ontVFXU\" 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=12779136&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gghgchfg-sBR_ontVFXU&vembed=0&video_id=sBR_ontVFXU&video_target=tpm-plugin-gghgchfg-sBR_ontVFXU'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Isolate+binomial+absolute+value_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIsolate binomial absolute value\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section aria-label=\"Watch It\">\r\n<h2>Absolute Value Inequalities<\/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 of Absolute Value Inequalities:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Equations of the form [latex]|A| &lt; B[\/latex], [latex]|A| \\leq B[\/latex], [latex]|A| &gt; B[\/latex], or [latex]|A| \\geq B[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]A[\/latex] and [latex]B[\/latex] are algebraic expressions, often involving a variable [latex]x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solving Absolute Value Inequalities:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For [latex]|X| &lt; k[\/latex] (where [latex]k &gt; 0[\/latex]): Equivalent to [latex]-k &lt; X &lt; k[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For [latex]|X| &gt; k[\/latex] (where [latex]k &gt; 0[\/latex]): Equivalent to [latex]X &lt; -k[\/latex] or [latex]X &gt; k[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Similar rules apply for [latex]\\leq[\/latex] and [latex]\\geq[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphical Interpretation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Solutions represent intervals on a number line<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]|X| &lt; k[\/latex]: Points within [latex]k[\/latex] units of zero<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]|X| &gt; k[\/latex]: Points more than [latex]k[\/latex] units away from zero<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Describe all [latex]x[\/latex]<em>-<\/em>values within a distance of [latex]3[\/latex] from the number [latex]2[\/latex].[reveal-answer q=\"7507\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"7507\"][latex]|x - 2|\\le 3[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<h3>Try It<\/h3>\r\nSolve [latex]-2|k - 4|\\le -6[\/latex].\r\n\r\n[reveal-answer q=\"96760\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"96760\"]\r\n\r\n[latex]k\\le 1[\/latex] or [latex]k\\ge 7[\/latex]; in interval notation, this would be [latex]\\left(-\\infty ,1\\right]\\cup \\left[7,\\infty \\right)[\/latex].\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200413\/CNX_CAT_Figure_02_07_007.jpg\" alt=\"A coordinate plane with the x-axis ranging from -1 to 9 and the y-axis ranging from -3 to 8. The function y = -2|k 4| + 6 is graphed and everything above the function is shaded in.\" \/>\r\n\r\n[\/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-ddghfgfa-b7dQJ19_D4E\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/b7dQJ19_D4E?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ddghfgfa-b7dQJ19_D4E\" 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=12779137&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ddghfgfa-b7dQJ19_D4E&vembed=0&video_id=b7dQJ19_D4E&video_target=tpm-plugin-ddghfgfa-b7dQJ19_D4E'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Absolute+Value+Inequality_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAbsolute Value Inequality\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Solve equations that include fractions with variables.<\/li>\n<li>Solve equations with roots and fractional powers.<\/li>\n<li>Use factoring to find solutions to polynomial equations.<\/li>\n<li>Find solutions to equations that involve absolute values.<\/li>\n<li><span data-sheets-root=\"1\">Find solutions to inequalities that involve absolute values.<\/span><\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Solving a Rational Equation<\/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 of Rational Equations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Contains at least one rational expression<\/li>\n<li class=\"whitespace-normal break-words\">Variable appears in at least one denominator<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solving Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Factor all denominators<\/li>\n<li class=\"whitespace-normal break-words\">Find and exclude values that make denominators zero<\/li>\n<li class=\"whitespace-normal break-words\">Determine the Least Common Denominator (LCD)<\/li>\n<li class=\"whitespace-normal break-words\">Multiply both sides by the LCD<\/li>\n<li class=\"whitespace-normal break-words\">Solve the resulting equation<\/li>\n<li class=\"whitespace-normal break-words\">Check solutions in the original equation<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Cross-Multiplication Method:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For equations in the form [latex]\\frac{a}{b} = \\frac{c}{d}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Multiply to get [latex]ad = bc[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Handling Binomial Denominators:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Treat binomials (e.g., [latex]x + 1[\/latex]) as single units<\/li>\n<li class=\"whitespace-normal break-words\">Factor completely before finding LCD<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Importance in Algebra:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Bridge between linear equations and more complex algebraic structures<\/li>\n<li class=\"whitespace-normal break-words\">Foundation for solving many real-world problems<strong>\u00a0<\/strong><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the rational equation:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{2}{3x} = \\dfrac{1}{4} - \\dfrac{1}{6x}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q407050\">Show Answer<\/button><\/p>\n<div id=\"q407050\" class=\"hidden-answer\" style=\"display: none\">[latex]x=\\dfrac{10}{3}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the rational equation:<\/p>\n<p style=\"text-align: center;\">[latex]-\\dfrac{5}{2x} + \\dfrac{3}{4x} = -\\dfrac{7}{4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q18903\">Show Answer<\/button><\/p>\n<div id=\"q18903\" class=\"hidden-answer\" style=\"display: none\">[latex]x=1[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]\\dfrac{-3}{2x+1} = \\dfrac{4}{3x+1}[\/latex]. State the excluded values.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q132629\">Show Answer<\/button><\/p>\n<div id=\"q132629\" class=\"hidden-answer\" style=\"display: none\">[latex]x=-\\dfrac{7}{17}[\/latex]. Excluded values are [latex]x= -\\dfrac{1}{2}[\/latex] and [latex]x= -\\dfrac{1}{3}[\/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-bbahfceh-UBfIQnTYGoY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/UBfIQnTYGoY?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bbahfceh-UBfIQnTYGoY\" 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=12844260&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-bbahfceh-UBfIQnTYGoY&#38;vembed=0&#38;video_id=UBfIQnTYGoY&#38;video_target=tpm-plugin-bbahfceh-UBfIQnTYGoY\"><\/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+2+-+Solving+Rational+Equations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Solving Rational Equations\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Radical Equations<\/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\">Equations containing variables under a radical symbol<\/li>\n<li class=\"whitespace-normal break-words\">Example: [latex]\\sqrt{3x + 18} = x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solving Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Isolate the radical term<\/li>\n<li class=\"whitespace-normal break-words\">Raise both sides to the power of the radical&#8217;s index<\/li>\n<li class=\"whitespace-normal break-words\">Solve the resulting equation<\/li>\n<li class=\"whitespace-normal break-words\">Check for extraneous solutions<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Extraneous Solutions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Solutions that satisfy the altered equation but not the original<\/li>\n<li class=\"whitespace-normal break-words\">Result from squaring or cubing both sides<\/li>\n<li class=\"whitespace-normal break-words\">Must be checked by substitution in the original equation<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Multiple Radicals:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Isolate one radical at a time<\/li>\n<li class=\"whitespace-normal break-words\">Repeat the process for each radical<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify all radical terms<\/li>\n<li class=\"whitespace-normal break-words\">Plan the isolation sequence for multiple radicals<\/li>\n<li class=\"whitespace-normal break-words\">Raise both sides to appropriate powers<\/li>\n<li class=\"whitespace-normal break-words\">Solve the resulting polynomial equation<\/li>\n<li class=\"whitespace-normal break-words\">Check all solutions in the original equation<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the radical equation: [latex]\\sqrt{x+3}=3x - 1[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q719648\">Show Solution<\/button><\/p>\n<div id=\"q719648\" class=\"hidden-answer\" style=\"display: none\">[latex]x=1[\/latex]; extraneous solution [latex]x=-\\frac{2}{9}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the equation with two radicals: [latex]\\sqrt{3x+7}+\\sqrt{x+2}=1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q265496\">Show Solution<\/button><\/p>\n<div id=\"q265496\" class=\"hidden-answer\" style=\"display: none\">[latex]x=-2[\/latex]; extraneous solution [latex]x=-1[\/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-dfeaehce-RK0lFf43gSY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/RK0lFf43gSY?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dfeaehce-RK0lFf43gSY\" 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=12844261&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-dfeaehce-RK0lFf43gSY&#38;vembed=0&#38;video_id=RK0lFf43gSY&#38;video_target=tpm-plugin-dfeaehce-RK0lFf43gSY\"><\/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+-+Solve+Radical+Equations+-+Square+Roots_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 3: Solve Radical Equations &#8211; Square Roots\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Solve Equations With Rational Exponents<\/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\">Rational Exponents:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Fractions as exponents<\/li>\n<li class=\"whitespace-normal break-words\">Notation: [latex]a^{\\frac{m}{n}} = \\sqrt[n]{a^m}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Equivalence to Radicals:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]a^{\\frac{1}{n}} = \\sqrt[n]{a}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a^{\\frac{m}{n}} = (\\sqrt[n]{a})^m[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solving Strategy:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Raise both sides to the reciprocal power<\/li>\n<li class=\"whitespace-normal break-words\">Simplify using exponent rules<\/li>\n<li class=\"whitespace-normal break-words\">Solve the resulting equation<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Exponent Rules:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Product: [latex]a^m \\cdot a^n = a^{m+n}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Quotient: [latex]\\frac{a^m}{a^n} = a^{m-n}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Power: [latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Technique<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the rational exponent<\/li>\n<li class=\"whitespace-normal break-words\">Determine its reciprocal<\/li>\n<li class=\"whitespace-normal break-words\">Apply the reciprocal exponent to both sides<\/li>\n<li class=\"whitespace-normal break-words\">Simplify using exponent rules<\/li>\n<li class=\"whitespace-normal break-words\">Solve the resulting equation<\/li>\n<li class=\"whitespace-normal break-words\">Check the solution<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate [latex]{64}^{-\\frac{1}{3}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q68783\">Show Solution<\/button><\/p>\n<div id=\"q68783\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{1}{4}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the equation [latex]{x}^{\\frac{3}{2}}=125[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q390459\">Show Solution<\/button><\/p>\n<div id=\"q390459\" class=\"hidden-answer\" style=\"display: none\">[latex]25[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve: [latex]{\\left(x+5\\right)}^{\\frac{3}{2}}=8[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q943422\">Show Solution<\/button><\/p>\n<div id=\"q943422\" class=\"hidden-answer\" style=\"display: none\">[latex]-1[\/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-hhdhceha-tS6J8Vb81RE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/tS6J8Vb81RE?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hhdhceha-tS6J8Vb81RE\" 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=12844262&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-hhdhceha-tS6J8Vb81RE&#38;vembed=0&#38;video_id=tS6J8Vb81RE&#38;video_target=tpm-plugin-hhdhceha-tS6J8Vb81RE\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Solve+Equations+with+Rational+Exponents+(Two+Solutions)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolve Equations with Rational Exponents (Two Solutions)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Polynomial Equations<\/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: A polynomial equation is an equation of the form: [latex]a_nx^n + a_{n-1}x^{n-1} + \\ldots + a_2x^2 + a_1x + a_0 = 0[\/latex] where [latex]n[\/latex] is a positive integer and [latex]a_n \\neq 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Degree:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The highest power of the variable in the polynomial<\/li>\n<li class=\"whitespace-normal break-words\">Determines the maximum number of solutions<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Zero-Product Property: If [latex]ab = 0[\/latex], then [latex]a = 0[\/latex] or [latex]b = 0[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Fundamental to solving polynomial equations by factoring<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solution Types:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Real solutions (rational or irrational)<\/li>\n<li class=\"whitespace-normal break-words\">Complex solutions (when real solutions don&#8217;t exist)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Steps<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Arrange the polynomial in standard form (descending powers)<\/li>\n<li class=\"whitespace-normal break-words\">Factor out the greatest common factor (GCF)<\/li>\n<li class=\"whitespace-normal break-words\">Look for special patterns or grouping opportunities<\/li>\n<li class=\"whitespace-normal break-words\">Factor completely<\/li>\n<li class=\"whitespace-normal break-words\">Apply the zero-product property<\/li>\n<li class=\"whitespace-normal break-words\">Solve the resulting linear equations<\/li>\n<li class=\"whitespace-normal break-words\">Check solutions in the original equation<\/li>\n<\/ol>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve:<\/p>\n<p style=\"text-align: center;\">[latex]2x^4 - 18x^2 + 40 = 0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q262759\">Show Answer<\/button><\/p>\n<div id=\"q262759\" class=\"hidden-answer\" style=\"display: none\">\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Factor out GCF: [latex]2(x^4 - 9x^2 + 20) = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Recognize [latex]x^4 - 9x^2 + 20[\/latex] as a quadratic in [latex]x^2[\/latex] Let [latex]u = x^2[\/latex], then [latex]u^2 - 9u + 20 = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Factor the quadratic in [latex]u[\/latex]: [latex]2(u - 4)(u - 5) = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Substitute back [latex]x^2[\/latex] for [latex]u[\/latex]: [latex]2(x^2 - 4)(x^2 - 5) = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Apply zero-product property: [latex]2 = 0[\/latex] (always false) [latex]x^2 - 4 = 0[\/latex] or [latex]x^2 - 5 = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Solve resulting equations: [latex]x = \\pm 2[\/latex] or [latex]x = \\pm \\sqrt{5}[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the solutions are [latex]x = 2, -2, \\sqrt{5}, -\\sqrt{5}[\/latex]<\/p>\n<\/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-debhabcg-5tiQN3wQZfs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/5tiQN3wQZfs?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-debhabcg-5tiQN3wQZfs\" 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=12844263&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-debhabcg-5tiQN3wQZfs&#38;vembed=0&#38;video_id=5tiQN3wQZfs&#38;video_target=tpm-plugin-debhabcg-5tiQN3wQZfs\"><\/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+-+Factor+and+Solve+a+Polynomial+Equation_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Factor and Solve a Polynomial Equation\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Absolute Value Equations<\/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 of Absolute Value:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Represents the distance of a number from zero on the number line<\/li>\n<li class=\"whitespace-normal break-words\">Always non-negative<\/li>\n<li class=\"whitespace-normal break-words\">Formally defined as: [latex]|x| = \\begin{cases} x & \\text{if } x \\geq 0 \\\\ -x & \\text{if } x < 0 \\end{cases}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Absolute Value Equation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">An equation containing an absolute value expression<\/li>\n<li class=\"whitespace-normal break-words\">General form: [latex]|A| = B[\/latex], where A is an expression and B is a non-negative number<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Properties of Absolute Value Equations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]|A| = B[\/latex], then [latex]A = B[\/latex] or [latex]A = -B[\/latex] when [latex]B \\geq 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]B < 0[\/latex], the equation [latex]|A| = B[\/latex] has no solution<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Standard Form of Linear Absolute Value Equations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]|ax + b| = c[\/latex], where [latex]a \\neq 0[\/latex] and [latex]c[\/latex] is a real number<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Number of Solutions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]c < 0[\/latex]: No solution<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]c = 0[\/latex]: One solution<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]c > 0[\/latex]: Two solutions<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>Solving Process:<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Isolate the absolute value expression on one side of the equation<\/li>\n<li class=\"whitespace-normal break-words\">Consider two cases: positive and negative<\/li>\n<li class=\"whitespace-normal break-words\">Solve each case as a linear equation<\/li>\n<li class=\"whitespace-normal break-words\">Check solutions in the original equation<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the absolute value equation: [latex]|1 - 4x|+8=13[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q567620\">Show Solution<\/button><\/p>\n<div id=\"q567620\" class=\"hidden-answer\" style=\"display: none\">[latex]x=-1[\/latex], [latex]x=\\frac{3}{2}[\/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-gghgchfg-sBR_ontVFXU\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/sBR_ontVFXU?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gghgchfg-sBR_ontVFXU\" 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=12779136&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-gghgchfg-sBR_ontVFXU&#38;vembed=0&#38;video_id=sBR_ontVFXU&#38;video_target=tpm-plugin-gghgchfg-sBR_ontVFXU\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Isolate+binomial+absolute+value_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIsolate binomial absolute value\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section aria-label=\"Watch It\">\n<h2>Absolute Value Inequalities<\/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 of Absolute Value Inequalities:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Equations of the form [latex]|A| < B[\/latex], [latex]|A| \\leq B[\/latex], [latex]|A| > B[\/latex], or [latex]|A| \\geq B[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]A[\/latex] and [latex]B[\/latex] are algebraic expressions, often involving a variable [latex]x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solving Absolute Value Inequalities:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]|X| < k[\/latex] (where [latex]k > 0[\/latex]): Equivalent to [latex]-k < X < k[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">For [latex]|X| > k[\/latex] (where [latex]k > 0[\/latex]): Equivalent to [latex]X < -k[\/latex] or [latex]X > k[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Similar rules apply for [latex]\\leq[\/latex] and [latex]\\geq[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graphical Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Solutions represent intervals on a number line<\/li>\n<li class=\"whitespace-normal break-words\">[latex]|X| < k[\/latex]: Points within [latex]k[\/latex] units of zero<\/li>\n<li class=\"whitespace-normal break-words\">[latex]|X| > k[\/latex]: Points more than [latex]k[\/latex] units away from zero<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Describe all [latex]x[\/latex]<em>&#8211;<\/em>values within a distance of [latex]3[\/latex] from the number [latex]2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q7507\">Show Solution<\/button><\/p>\n<div id=\"q7507\" class=\"hidden-answer\" style=\"display: none\">[latex]|x - 2|\\le 3[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<h3>Try It<\/h3>\n<p>Solve [latex]-2|k - 4|\\le -6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q96760\">Show Solution<\/button><\/p>\n<div id=\"q96760\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]k\\le 1[\/latex] or [latex]k\\ge 7[\/latex]; in interval notation, this would be [latex]\\left(-\\infty ,1\\right]\\cup \\left[7,\\infty \\right)[\/latex].<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200413\/CNX_CAT_Figure_02_07_007.jpg\" alt=\"A coordinate plane with the x-axis ranging from -1 to 9 and the y-axis ranging from -3 to 8. The function y = -2|k 4| + 6 is graphed and everything above the function is shaded in.\" \/><\/p>\n<\/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-ddghfgfa-b7dQJ19_D4E\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/b7dQJ19_D4E?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ddghfgfa-b7dQJ19_D4E\" 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=12779137&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ddghfgfa-b7dQJ19_D4E&#38;vembed=0&#38;video_id=b7dQJ19_D4E&#38;video_target=tpm-plugin-ddghfgfa-b7dQJ19_D4E\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Absolute+Value+Inequality_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAbsolute Value Inequality\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/section>\n","protected":false},"author":12,"menu_order":18,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex 2: Solving Rational Equations \",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/www.youtube.com\/watch?v=UBfIQnTYGoY\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 3: Solve Radical Equations - Square Roots \",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/www.youtube.com\/watch?v=RK0lFf43gSY\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Solve Equations with Rational Exponents (Two Solutions) \",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/www.youtube.com\/watch?v=tS6J8Vb81RE\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Factor and Solve a Polynomial Equation 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