{"id":1411,"date":"2025-07-25T01:05:35","date_gmt":"2025-07-25T01:05:35","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1411"},"modified":"2026-03-11T08:58:56","modified_gmt":"2026-03-11T08:58:56","slug":"polynomial-equations-and-inequalities-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polynomial-equations-and-inequalities-fresh-take\/","title":{"raw":"Polynomial Equations and Inequalities: Fresh Take","rendered":"Polynomial Equations and Inequalities: Fresh Take"},"content":{"raw":"<h2>Solving Real-World Applications of Polynomial Equations<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\nPolynomial equations show up everywhere in the real world\u2014from calculating volumes and areas to modeling business profits and projectile motion. The key is translating the word problem into a polynomial equation, solving it, and then interpreting your answer in context.Remember, not all mathematical solutions make sense in real-world contexts. A negative length or a time before the experiment started might solve your equation but won't answer the actual question!<\/div>\r\n<section class=\"textbox example\"><strong>Example:<\/strong> Volume of a Cone\r\nA mound of gravel is shaped like a cone with height equal to twice the radius. The volume formula is [latex]V = \\frac{2}{3}\\pi r^3[\/latex]. If a customer purchases 100 cubic feet of gravel, what are the radius and height of the cone?\r\n[reveal-answer q=\"poly1\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"poly1\"]\r\n[latex]\\begin{align}\r\n\\text{Start with volume formula} &amp;: V = \\frac{2}{3}\\pi r^3 \\\r\n\\text{Substitute } V = 100 &amp;: 100 = \\frac{2}{3}\\pi r^3 \\\r\n\\text{Multiply both sides by } \\frac{3}{2\\pi} &amp;: \\frac{300}{2\\pi} = r^3 \\\r\n\\text{Simplify} &amp;: \\frac{150}{\\pi} = r^3 \\\r\n\\text{Take cube root} &amp;: r = \\sqrt[3]{\\frac{150}{\\pi}} \\\r\n\\text{Calculate} &amp;: r \\approx 3.63 \\text{ feet}\r\n\\end{align}[\/latex]\r\nSince height is twice the radius: [latex]h = 2(3.63) \\approx 7.26[\/latex] feet\r\nThe cone has a radius of approximately 3.63 feet and a height of approximately 7.26 feet.\r\n[\/hidden-answer]<\/section>\r\n<div class=\"textbox proTip\"><strong>Pro Tip:<\/strong> Always check if your answer makes sense in the real-world context. Negative distances, impossible times, or values outside reasonable bounds are clues you need to reconsider your solution.<\/div>\r\n<section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]317806[\/ohm_question]<\/section>\r\n<h2>Solving Polynomial Inequalities<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\nPolynomial inequalities ask \"when is this polynomial positive?\" or \"when is it negative?\"The key insight: polynomials only change sign at their zeros (where they cross the x-axis). So we find the zeros, then test the intervals between them.<\/div>\r\n<div class=\"textbox questionHelp\">\r\n\r\n<strong>Question Help: Solving Polynomial Inequalities<\/strong>\r\n<ol>\r\n \t<li>Solve the related equation (set the polynomial equal to zero) to find the zeros.<\/li>\r\n \t<li>Plot these zeros on a number line\u2014they divide the line into intervals.<\/li>\r\n \t<li>Choose a test value in each interval.<\/li>\r\n \t<li>Evaluate the polynomial at each test value to determine if it's positive or negative in that interval.<\/li>\r\n \t<li>Select the intervals that satisfy your inequality.<\/li>\r\n \t<li>Write your answer in interval notation.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\r\n<h3>Solving a polynomial inequality not in factored form - use greatest common factor.<\/h3>\r\n<h3><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script><\/h3>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ggadceaf-zyiad-T6-TI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/zyiad-T6-TI?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ggadceaf-zyiad-T6-TI\" 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=14660186&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ggadceaf-zyiad-T6-TI&vembed=0&video_id=zyiad-T6-TI&video_target=tpm-plugin-ggadceaf-zyiad-T6-TI'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Ex+-+Solve+a+Polynomial+Inequality+-+Factor+Using+GCF+(Degree+3)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve a Polynomial Inequality - Factor Using GCF (Degree 3)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">\r\n<h3>Solving a polynomial inequality not in factored form - factor a trinomial<\/h3>\r\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-acddcacg-LC1bwRHcdh4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/LC1bwRHcdh4?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-acddcacg-LC1bwRHcdh4\" 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=14660187&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-acddcacg-LC1bwRHcdh4&vembed=0&video_id=LC1bwRHcdh4&video_target=tpm-plugin-acddcacg-LC1bwRHcdh4'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Ex+-+Solve+a+Polynomial+Inequality+-+Factor+a+Trinomial+(Degree+4)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve a Polynomial Inequality - Factor a Trinomial (Degree 4)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\"><strong>Example:<\/strong> Solve [latex](x + 3)(x + 1)^2(x - 4) &gt; 0[\/latex]\r\n[reveal-answer q=\"poly2\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"poly2\"]\r\nFirst, find the zeros by solving [latex](x + 3)(x + 1)^2(x - 4) = 0[\/latex]:\r\n<ul>\r\n \t<li>[latex]x = -3[\/latex], [latex]x = -1[\/latex], and [latex]x = 4[\/latex]<\/li>\r\n<\/ul>\r\nThese divide the number line into 4 intervals. Test a value in each:\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Interval<\/th>\r\n<th>Test Value<\/th>\r\n<th>Sign of [latex]f(x)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]x &lt; -3[\/latex]<\/td>\r\n<td>[latex]x = -4[\/latex]<\/td>\r\n<td>[latex]f(-4) = 72 &gt; 0[\/latex] \u2713<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3 &lt; x &lt; -1[\/latex]<\/td>\r\n<td>[latex]x = -2[\/latex]<\/td>\r\n<td>[latex]f(-2) = -6 &lt; 0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1 &lt; x &lt; 4[\/latex]<\/td>\r\n<td>[latex]x = 0[\/latex]<\/td>\r\n<td>[latex]f(0) = -12 &lt; 0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x &gt; 4[\/latex]<\/td>\r\n<td>[latex]x = 5[\/latex]<\/td>\r\n<td>[latex]f(5) = 288 &gt; 0[\/latex] \u2713<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe function is positive when [latex]x &lt; -3[\/latex] or [latex]x &gt; 4[\/latex].\r\nSolution: [latex](-\\infty, -3) \\cup (4, \\infty)[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<div class=\"textbox recall\"><strong>Recall:<\/strong> Use parentheses [latex]( )[\/latex] for [latex]&lt;[\/latex] or [latex]&gt;[\/latex], and brackets [latex][ ][\/latex] for [latex]\\leq[\/latex] or [latex]\\geq[\/latex]. The union symbol [latex]\\cup[\/latex] combines separate intervals.<\/div>\r\n<section class=\"textbox example\"><strong>Example:<\/strong> Find the domain of [latex]v(t) = \\sqrt{6 - 5t - t^2}[\/latex]\r\n[reveal-answer q=\"poly3\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"poly3\"]\r\nA square root is only defined when the expression inside is non-negative, so we need [latex]6 - 5t - t^2 \\geq 0[\/latex].\r\n[latex]\\begin{align}\r\n\\text{Solve the equation} &amp;: 6 - 5t - t^2 = 0 \\\r\n\\text{Factor} &amp;: (6 + t)(1 - t) = 0 \\\r\n\\text{Zeros} &amp;: t = -6 \\text{ and } t = 1\r\n\\end{align}[\/latex]\r\nThis is a downward-opening parabola (negative [latex]t^2[\/latex] term), so it's positive between its zeros.\r\nThe domain is [latex][-6, 1][\/latex].\r\n[\/hidden-answer]<\/section>\r\n<div class=\"textbox proTip\"><strong>Pro Tip:<\/strong> For inequalities involving square roots or fractions, the polynomial inequality tells you where the function is defined (the domain).<\/div>\r\n<section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]317805[\/ohm_question]<\/section>\r\n<h2>Finding the Inverse of Invertible Polynomial Functions<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\nAn inverse function \"undoes\" what the original function does. If [latex]f(x) = 5[\/latex] when [latex]x = 2[\/latex], then [latex]f^{-1}(5) = 2[\/latex]. The inputs and outputs swap places!However, only one-to-one functions have inverses that are also functions. Most polynomials aren't one-to-one, but some simple ones (like odd-degree polynomials) are.<strong>Warning:<\/strong> [latex]f^{-1}(x)[\/latex] does NOT mean [latex]\\frac{1}{f(x)}[\/latex]! The notation [latex]f^{-1}[\/latex] specifically means \"inverse function,\" not reciprocal.<\/div>\r\n<div class=\"textbox questionHelp\">\r\n\r\n<strong>Question Help: Finding an Inverse Function<\/strong>\r\n<ol>\r\n \t<li>Verify the function is one-to-one (passes the horizontal line test).<\/li>\r\n \t<li>Replace [latex]f(x)[\/latex] with [latex]y[\/latex].<\/li>\r\n \t<li>Swap [latex]x[\/latex] and [latex]y[\/latex].<\/li>\r\n \t<li>Solve for [latex]y[\/latex].<\/li>\r\n \t<li>Replace [latex]y[\/latex] with [latex]f^{-1}(x)[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\"><strong>Example:<\/strong> Find the inverse of [latex]f(x) = 5x^3 + 1[\/latex]\r\n[reveal-answer q=\"poly4\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"poly4\"]\r\n[latex]\\begin{align}\r\n\\text{Replace } f(x) \\text{ with } y &amp;: y = 5x^3 + 1 \\\r\n\\text{Swap } x \\text{ and } y &amp;: x = 5y^3 + 1 \\\r\n\\text{Subtract 1} &amp;: x - 1 = 5y^3 \\\r\n\\text{Divide by 5} &amp;: \\frac{x - 1}{5} = y^3 \\\r\n\\text{Take cube root} &amp;: y = \\sqrt[3]{\\frac{x - 1}{5}} \\\r\n\\text{Write inverse} &amp;: f^{-1}(x) = \\sqrt[3]{\\frac{x - 1}{5}}\r\n\\end{align}[\/latex]\r\n[\/hidden-answer]<\/section>\r\n<div class=\"textbox recall\"><strong>Recall:<\/strong> The graphs of [latex]f[\/latex] and [latex]f^{-1}[\/latex] are reflections of each other across the line [latex]y = x[\/latex]. If [latex](a, b)[\/latex] is on [latex]f[\/latex], then [latex](b, a)[\/latex] is on [latex]f^{-1}[\/latex].<\/div>\r\n<div class=\"textbox proTip\"><strong>Pro Tip:<\/strong> Check your work by composing the functions: [latex]f(f^{-1}(x)) = x[\/latex] and [latex]f^{-1}(f(x)) = x[\/latex] should both be true.<\/div>\r\n<h2>Restricting the Domain to Find Inverses<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\nMost polynomial functions aren't one-to-one over their entire domain. A parabola, for example, fails the horizontal line test.But if we restrict the domain (only use part of the graph), we can create a one-to-one function that does have an inverse.Think of it like only walking up one side of a hill\u2014that way, each height corresponds to exactly one location on your path.<\/div>\r\n<div class=\"textbox questionHelp\">\r\n\r\n<strong>Question Help: Restricting Domain to Find an Inverse<\/strong>\r\n<ol>\r\n \t<li>Identify where the function is one-to-one (often one side of the vertex for quadratics).<\/li>\r\n \t<li>Restrict the domain to that interval.<\/li>\r\n \t<li>Replace [latex]f(x)[\/latex] with [latex]y[\/latex].<\/li>\r\n \t<li>Swap [latex]x[\/latex] and [latex]y[\/latex].<\/li>\r\n \t<li>Solve for [latex]y[\/latex]\u2014you may get [latex]\\pm[\/latex] from a square root.<\/li>\r\n \t<li>Choose the sign ([latex]+[\/latex] or [latex]-[\/latex]) that matches your restricted domain.<\/li>\r\n \t<li>Write the inverse with its domain restriction.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\"><strong>Example:<\/strong> Find the inverse of [latex]f(x) = (x - 4)^2[\/latex] with domain [latex]x \\geq 4[\/latex]\r\n[reveal-answer q=\"poly5\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"poly5\"]\r\n[latex]\\begin{align}\r\n\\text{Replace } f(x) \\text{ with } y &amp;: y = (x - 4)^2 \\\r\n\\text{Swap } x \\text{ and } y &amp;: x = (y - 4)^2 \\\r\n\\text{Take square root} &amp;: \\pm\\sqrt{x} = y - 4 \\\r\n\\text{Add 4} &amp;: 4 \\pm \\sqrt{x} = y\r\n\\end{align}[\/latex]\r\nSince the original function has domain [latex]x \\geq 4[\/latex], the inverse must have range [latex]y \\geq 4[\/latex]. We use the positive sign:\r\n[latex]f^{-1}(x) = 4 + \\sqrt{x}[\/latex]\r\n[\/hidden-answer]<\/section><section class=\"textbox example\"><strong>Example:<\/strong> Restrict the domain and find the inverse of [latex]f(x) = (x - 2)^2 - 3[\/latex]\r\n[reveal-answer q=\"poly6\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"poly6\"]\r\nThis is a parabola with vertex at [latex](2, -3)[\/latex] opening upward. We can restrict to [latex]x \\geq 2[\/latex] (the right side).\r\n[latex]\\begin{align}\r\n\\text{Replace } f(x) \\text{ with } y &amp;: y = (x - 2)^2 - 3 \\\r\n\\text{Swap } x \\text{ and } y &amp;: x = (y - 2)^2 - 3 \\\r\n\\text{Add 3} &amp;: x + 3 = (y - 2)^2 \\\r\n\\text{Take square root} &amp;: \\pm\\sqrt{x + 3} = y - 2 \\\r\n\\text{Add 2} &amp;: 2 \\pm \\sqrt{x + 3} = y\r\n\\end{align}[\/latex]\r\nSince we restricted to [latex]x \\geq 2[\/latex], we need [latex]y \\geq 2[\/latex]:\r\n[latex]f^{-1}(x) = 2 + \\sqrt{x + 3}[\/latex]\r\nNote: If we had restricted to [latex]x \\leq 2[\/latex] instead, the inverse would be [latex]f^{-1}(x) = 2 - \\sqrt{x + 3}[\/latex].\r\n[\/hidden-answer]<\/section>\r\n<div class=\"textbox proTip\"><strong>Pro Tip:<\/strong> For quadratics in vertex form, the vertex tells you where to split the domain. Choose one side of the vertex to restrict to.<\/div>\r\n<div class=\"textbox recall\"><strong>Recall:<\/strong> The domain of [latex]f[\/latex] becomes the range of [latex]f^{-1}[\/latex], and the range of [latex]f[\/latex] becomes the domain of [latex]f^{-1}[\/latex]. They swap!<\/div>","rendered":"<h2>Solving Real-World Applications of Polynomial Equations<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><br \/>\nPolynomial equations show up everywhere in the real world\u2014from calculating volumes and areas to modeling business profits and projectile motion. The key is translating the word problem into a polynomial equation, solving it, and then interpreting your answer in context.Remember, not all mathematical solutions make sense in real-world contexts. A negative length or a time before the experiment started might solve your equation but won&#8217;t answer the actual question!<\/div>\n<section class=\"textbox example\"><strong>Example:<\/strong> Volume of a Cone<br \/>\nA mound of gravel is shaped like a cone with height equal to twice the radius. The volume formula is [latex]V = \\frac{2}{3}\\pi r^3[\/latex]. If a customer purchases 100 cubic feet of gravel, what are the radius and height of the cone?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qpoly1\">Show Solution<\/button><\/p>\n<div id=\"qpoly1\" class=\"hidden-answer\" style=\"display: none\">\n[latex]\\begin{align}  \\text{Start with volume formula} &: V = \\frac{2}{3}\\pi r^3 \\  \\text{Substitute } V = 100 &: 100 = \\frac{2}{3}\\pi r^3 \\  \\text{Multiply both sides by } \\frac{3}{2\\pi} &: \\frac{300}{2\\pi} = r^3 \\  \\text{Simplify} &: \\frac{150}{\\pi} = r^3 \\  \\text{Take cube root} &: r = \\sqrt[3]{\\frac{150}{\\pi}} \\  \\text{Calculate} &: r \\approx 3.63 \\text{ feet}  \\end{align}[\/latex]<br \/>\nSince height is twice the radius: [latex]h = 2(3.63) \\approx 7.26[\/latex] feet<br \/>\nThe cone has a radius of approximately 3.63 feet and a height of approximately 7.26 feet.\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox proTip\"><strong>Pro Tip:<\/strong> Always check if your answer makes sense in the real-world context. Negative distances, impossible times, or values outside reasonable bounds are clues you need to reconsider your solution.<\/div>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm317806\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317806&theme=lumen&iframe_resize_id=ohm317806&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Solving Polynomial Inequalities<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><br \/>\nPolynomial inequalities ask &#8220;when is this polynomial positive?&#8221; or &#8220;when is it negative?&#8221;The key insight: polynomials only change sign at their zeros (where they cross the x-axis). So we find the zeros, then test the intervals between them.<\/div>\n<div class=\"textbox questionHelp\">\n<p><strong>Question Help: Solving Polynomial Inequalities<\/strong><\/p>\n<ol>\n<li>Solve the related equation (set the polynomial equal to zero) to find the zeros.<\/li>\n<li>Plot these zeros on a number line\u2014they divide the line into intervals.<\/li>\n<li>Choose a test value in each interval.<\/li>\n<li>Evaluate the polynomial at each test value to determine if it&#8217;s positive or negative in that interval.<\/li>\n<li>Select the intervals that satisfy your inequality.<\/li>\n<li>Write your answer in interval notation.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\n<h3>Solving a polynomial inequality not in factored form &#8211; use greatest common factor.<\/h3>\n<h3><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/h3>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ggadceaf-zyiad-T6-TI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/zyiad-T6-TI?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ggadceaf-zyiad-T6-TI\" 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=14660186&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ggadceaf-zyiad-T6-TI&#38;vembed=0&#38;video_id=zyiad-T6-TI&#38;video_target=tpm-plugin-ggadceaf-zyiad-T6-TI\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Ex+-+Solve+a+Polynomial+Inequality+-+Factor+Using+GCF+(Degree+3)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve a Polynomial Inequality &#8211; Factor Using GCF (Degree 3)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\n<h3>Solving a polynomial inequality not in factored form &#8211; factor a trinomial<\/h3>\n<p><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-acddcacg-LC1bwRHcdh4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/LC1bwRHcdh4?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-acddcacg-LC1bwRHcdh4\" 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=14660187&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-acddcacg-LC1bwRHcdh4&#38;vembed=0&#38;video_id=LC1bwRHcdh4&#38;video_target=tpm-plugin-acddcacg-LC1bwRHcdh4\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Ex+-+Solve+a+Polynomial+Inequality+-+Factor+a+Trinomial+(Degree+4)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve a Polynomial Inequality &#8211; Factor a Trinomial (Degree 4)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\"><strong>Example:<\/strong> Solve [latex](x + 3)(x + 1)^2(x - 4) > 0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qpoly2\">Show Solution<\/button><\/p>\n<div id=\"qpoly2\" class=\"hidden-answer\" style=\"display: none\">\nFirst, find the zeros by solving [latex](x + 3)(x + 1)^2(x - 4) = 0[\/latex]:<\/p>\n<ul>\n<li>[latex]x = -3[\/latex], [latex]x = -1[\/latex], and [latex]x = 4[\/latex]<\/li>\n<\/ul>\n<p>These divide the number line into 4 intervals. Test a value in each:<\/p>\n<table>\n<thead>\n<tr>\n<th>Interval<\/th>\n<th>Test Value<\/th>\n<th>Sign of [latex]f(x)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]x < -3[\/latex]<\/td>\n<td>[latex]x = -4[\/latex]<\/td>\n<td>[latex]f(-4) = 72 > 0[\/latex] \u2713<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3 < x < -1[\/latex]<\/td>\n<td>[latex]x = -2[\/latex]<\/td>\n<td>[latex]f(-2) = -6 < 0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1 < x < 4[\/latex]<\/td>\n<td>[latex]x = 0[\/latex]<\/td>\n<td>[latex]f(0) = -12 < 0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]x > 4[\/latex]<\/td>\n<td>[latex]x = 5[\/latex]<\/td>\n<td>[latex]f(5) = 288 > 0[\/latex] \u2713<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The function is positive when [latex]x < -3[\/latex] or [latex]x > 4[\/latex].<br \/>\nSolution: [latex](-\\infty, -3) \\cup (4, \\infty)[\/latex]\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox recall\"><strong>Recall:<\/strong> Use parentheses [latex]( )[\/latex] for [latex]<[\/latex] or [latex]>[\/latex], and brackets [latex][ ][\/latex] for [latex]\\leq[\/latex] or [latex]\\geq[\/latex]. The union symbol [latex]\\cup[\/latex] combines separate intervals.<\/div>\n<section class=\"textbox example\"><strong>Example:<\/strong> Find the domain of [latex]v(t) = \\sqrt{6 - 5t - t^2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qpoly3\">Show Solution<\/button><\/p>\n<div id=\"qpoly3\" class=\"hidden-answer\" style=\"display: none\">\nA square root is only defined when the expression inside is non-negative, so we need [latex]6 - 5t - t^2 \\geq 0[\/latex].<br \/>\n[latex]\\begin{align}  \\text{Solve the equation} &: 6 - 5t - t^2 = 0 \\  \\text{Factor} &: (6 + t)(1 - t) = 0 \\  \\text{Zeros} &: t = -6 \\text{ and } t = 1  \\end{align}[\/latex]<br \/>\nThis is a downward-opening parabola (negative [latex]t^2[\/latex] term), so it&#8217;s positive between its zeros.<br \/>\nThe domain is [latex][-6, 1][\/latex].\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox proTip\"><strong>Pro Tip:<\/strong> For inequalities involving square roots or fractions, the polynomial inequality tells you where the function is defined (the domain).<\/div>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm317805\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317805&theme=lumen&iframe_resize_id=ohm317805&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Finding the Inverse of Invertible Polynomial Functions<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><br \/>\nAn inverse function &#8220;undoes&#8221; what the original function does. If [latex]f(x) = 5[\/latex] when [latex]x = 2[\/latex], then [latex]f^{-1}(5) = 2[\/latex]. The inputs and outputs swap places!However, only one-to-one functions have inverses that are also functions. Most polynomials aren&#8217;t one-to-one, but some simple ones (like odd-degree polynomials) are.<strong>Warning:<\/strong> [latex]f^{-1}(x)[\/latex] does NOT mean [latex]\\frac{1}{f(x)}[\/latex]! The notation [latex]f^{-1}[\/latex] specifically means &#8220;inverse function,&#8221; not reciprocal.<\/div>\n<div class=\"textbox questionHelp\">\n<p><strong>Question Help: Finding an Inverse Function<\/strong><\/p>\n<ol>\n<li>Verify the function is one-to-one (passes the horizontal line test).<\/li>\n<li>Replace [latex]f(x)[\/latex] with [latex]y[\/latex].<\/li>\n<li>Swap [latex]x[\/latex] and [latex]y[\/latex].<\/li>\n<li>Solve for [latex]y[\/latex].<\/li>\n<li>Replace [latex]y[\/latex] with [latex]f^{-1}(x)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\"><strong>Example:<\/strong> Find the inverse of [latex]f(x) = 5x^3 + 1[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qpoly4\">Show Solution<\/button><\/p>\n<div id=\"qpoly4\" class=\"hidden-answer\" style=\"display: none\">\n[latex]\\begin{align}  \\text{Replace } f(x) \\text{ with } y &: y = 5x^3 + 1 \\  \\text{Swap } x \\text{ and } y &: x = 5y^3 + 1 \\  \\text{Subtract 1} &: x - 1 = 5y^3 \\  \\text{Divide by 5} &: \\frac{x - 1}{5} = y^3 \\  \\text{Take cube root} &: y = \\sqrt[3]{\\frac{x - 1}{5}} \\  \\text{Write inverse} &: f^{-1}(x) = \\sqrt[3]{\\frac{x - 1}{5}}  \\end{align}[\/latex]\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox recall\"><strong>Recall:<\/strong> The graphs of [latex]f[\/latex] and [latex]f^{-1}[\/latex] are reflections of each other across the line [latex]y = x[\/latex]. If [latex](a, b)[\/latex] is on [latex]f[\/latex], then [latex](b, a)[\/latex] is on [latex]f^{-1}[\/latex].<\/div>\n<div class=\"textbox proTip\"><strong>Pro Tip:<\/strong> Check your work by composing the functions: [latex]f(f^{-1}(x)) = x[\/latex] and [latex]f^{-1}(f(x)) = x[\/latex] should both be true.<\/div>\n<h2>Restricting the Domain to Find Inverses<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><br \/>\nMost polynomial functions aren&#8217;t one-to-one over their entire domain. A parabola, for example, fails the horizontal line test.But if we restrict the domain (only use part of the graph), we can create a one-to-one function that does have an inverse.Think of it like only walking up one side of a hill\u2014that way, each height corresponds to exactly one location on your path.<\/div>\n<div class=\"textbox questionHelp\">\n<p><strong>Question Help: Restricting Domain to Find an Inverse<\/strong><\/p>\n<ol>\n<li>Identify where the function is one-to-one (often one side of the vertex for quadratics).<\/li>\n<li>Restrict the domain to that interval.<\/li>\n<li>Replace [latex]f(x)[\/latex] with [latex]y[\/latex].<\/li>\n<li>Swap [latex]x[\/latex] and [latex]y[\/latex].<\/li>\n<li>Solve for [latex]y[\/latex]\u2014you may get [latex]\\pm[\/latex] from a square root.<\/li>\n<li>Choose the sign ([latex]+[\/latex] or [latex]-[\/latex]) that matches your restricted domain.<\/li>\n<li>Write the inverse with its domain restriction.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\"><strong>Example:<\/strong> Find the inverse of [latex]f(x) = (x - 4)^2[\/latex] with domain [latex]x \\geq 4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qpoly5\">Show Solution<\/button><\/p>\n<div id=\"qpoly5\" class=\"hidden-answer\" style=\"display: none\">\n[latex]\\begin{align}  \\text{Replace } f(x) \\text{ with } y &: y = (x - 4)^2 \\  \\text{Swap } x \\text{ and } y &: x = (y - 4)^2 \\  \\text{Take square root} &: \\pm\\sqrt{x} = y - 4 \\  \\text{Add 4} &: 4 \\pm \\sqrt{x} = y  \\end{align}[\/latex]<br \/>\nSince the original function has domain [latex]x \\geq 4[\/latex], the inverse must have range [latex]y \\geq 4[\/latex]. We use the positive sign:<br \/>\n[latex]f^{-1}(x) = 4 + \\sqrt{x}[\/latex]\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\"><strong>Example:<\/strong> Restrict the domain and find the inverse of [latex]f(x) = (x - 2)^2 - 3[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qpoly6\">Show Solution<\/button><\/p>\n<div id=\"qpoly6\" class=\"hidden-answer\" style=\"display: none\">\nThis is a parabola with vertex at [latex](2, -3)[\/latex] opening upward. We can restrict to [latex]x \\geq 2[\/latex] (the right side).<br \/>\n[latex]\\begin{align}  \\text{Replace } f(x) \\text{ with } y &: y = (x - 2)^2 - 3 \\  \\text{Swap } x \\text{ and } y &: x = (y - 2)^2 - 3 \\  \\text{Add 3} &: x + 3 = (y - 2)^2 \\  \\text{Take square root} &: \\pm\\sqrt{x + 3} = y - 2 \\  \\text{Add 2} &: 2 \\pm \\sqrt{x + 3} = y  \\end{align}[\/latex]<br \/>\nSince we restricted to [latex]x \\geq 2[\/latex], we need [latex]y \\geq 2[\/latex]:<br \/>\n[latex]f^{-1}(x) = 2 + \\sqrt{x + 3}[\/latex]<br \/>\nNote: If we had restricted to [latex]x \\leq 2[\/latex] instead, the inverse would be [latex]f^{-1}(x) = 2 - \\sqrt{x + 3}[\/latex].\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox proTip\"><strong>Pro Tip:<\/strong> For quadratics in vertex form, the vertex tells you where to split the domain. Choose one side of the vertex to restrict to.<\/div>\n<div class=\"textbox recall\"><strong>Recall:<\/strong> The domain of [latex]f[\/latex] becomes the range of [latex]f^{-1}[\/latex], and the range of [latex]f[\/latex] becomes the domain of [latex]f^{-1}[\/latex]. They swap!<\/div>\n","protected":false},"author":67,"menu_order":27,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex: Solve a Polynomial Inequality - Factor Using GCF (Degree 3)\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/zyiad-T6-TI\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Solve a Polynomial Inequality - Factor a Trinomial (Degree 4)\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/LC1bwRHcdh4\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":506,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Ex: Solve a Polynomial Inequality - Factor Using GCF (Degree 3)","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/zyiad-T6-TI","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex: Solve a Polynomial Inequality - Factor a Trinomial (Degree 4)","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/LC1bwRHcdh4","project":"","license":"arr","license_terms":"Standard YouTube License"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><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=14660186&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ggadceaf-zyiad-T6-TI&vembed=0&video_id=zyiad-T6-TI&video_target=tpm-plugin-ggadceaf-zyiad-T6-TI'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><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=14660187&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-acddcacg-LC1bwRHcdh4&vembed=0&video_id=LC1bwRHcdh4&video_target=tpm-plugin-acddcacg-LC1bwRHcdh4'><\/script>\n","media_targets":["tpm-plugin-ggadceaf-zyiad-T6-TI","tpm-plugin-acddcacg-LC1bwRHcdh4"]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1411"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":12,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1411\/revisions"}],"predecessor-version":[{"id":5753,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1411\/revisions\/5753"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/506"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1411\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1411"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1411"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1411"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1411"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}