{"id":1408,"date":"2025-07-25T01:04:00","date_gmt":"2025-07-25T01:04:00","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1408"},"modified":"2026-03-18T05:40:29","modified_gmt":"2026-03-18T05:40:29","slug":"zeros-of-polynomial-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/zeros-of-polynomial-functions-fresh-take\/","title":{"raw":"Zeros of Polynomial Functions: Fresh Take","rendered":"Zeros of Polynomial Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Evaluate a polynomial using the Remainder Theorem.<\/li>\r\n \t<li>Use the Factor Theorem to solve a polynomial equation.<\/li>\r\n \t<li>Use the Rational Zero Theorem to find rational zeros.<\/li>\r\n \t<li>Find zeros of a polynomial function.<\/li>\r\n \t<li>Use the Linear Factorization Theorem to find polynomials with given zeros.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-edhehhea-D_I11k2DfCg\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/D_I11k2DfCg?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-edhehhea-D_I11k2DfCg\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12847030&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-edhehhea-D_I11k2DfCg&amp;vembed=0&amp;video_id=D_I11k2DfCg&amp;video_target=tpm-plugin-edhehhea-D_I11k2DfCg\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+2+-+Find+the+Zeros+of+a+Polynomial+Function+-+Real+Rational+Zeros_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Find the Zeros of a Polynomial Function - Real Rational Zeros\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Rational Zero Theorem<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\nWhen we're navigating the complex sea of polynomial functions, the Rational Zero Theorem is like our compass. It helps us pinpoint potential rational zeros using a simple yet powerful strategy: looking at the factors of the constant term and the leading coefficient.\r\n\r\nFor instance, consider a polynomial function that has zeros at [latex]\\frac{2}{5}[\/latex] and [latex]\\frac{3}{4}[\/latex]. These aren't just numbers; they're clues. By setting up equations with these zeros and constructing a quadratic function, we can see a pattern emerge. The numerators of these zeros ([latex]2[\/latex] and [latex]3[\/latex]) are factors of the constant term, while the denominators ([latex]5[\/latex] and [latex]4[\/latex]) are factors of the leading coefficient.\r\n\r\n<strong>Quick Tips: Applying the Rational Zero Theorem<\/strong>\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li><strong>List Out Factors<\/strong>: Start by listing all factors of the constant term and the leading coefficient of your polynomial.<\/li>\r\n \t<li><strong>Form Ratios<\/strong>: Create all possible ratios [latex]\\frac{p}{q}[\/latex] where [latex]p[\/latex] is a factor of the constant term and [latex]q[\/latex] is a factor of the leading coefficient.<\/li>\r\n \t<li><strong>Test Your Candidates<\/strong>: Evaluate each potential zero by plugging it into the polynomial. If the result is zero, you've found a true zero!<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Use the Rational Zero Theorem to find the rational zeros of [latex]f(x) = 2x^3 + x^2 - 4x + 1[\/latex].[reveal-answer q=\"981243\"]Show Solution[\/reveal-answer] [hidden-answer a=\"981243\"]The Rational Zero Theorem tells us that if [latex]\\frac{p}{q}[\/latex] is a zero of [latex]f(x)[\/latex], then [latex]p[\/latex] is a factor of [latex]1[\/latex] and [latex]q[\/latex] is a factor of [latex]2[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c c} \\frac{p}{q} &amp; = \\frac{\\text{factor of constant term}}{\\text{factor of leading coefficient}} \\\\ &amp; = \\frac{\\text{factor of 1}}{\\text{factor of 2}} \\end{array} [\/latex]<\/p>\r\nThe factors of [latex]1[\/latex] are [latex]\\pm1[\/latex] and the factors of [latex]2[\/latex] are [latex]\\pm1[\/latex] and [latex]\\pm2[\/latex]. The possible values for [latex]\\frac{p}{q}[\/latex] are [latex]\\pm1[\/latex] and [latex]\\pm\\frac{1}{2}[\/latex]. These are the possible rational zeros for the function. We can determine which of the possible zeros are actual zeros by substituting these values for [latex]x[\/latex] in [latex]f(x)[\/latex].\r\n<p style=\"text-align: center;\">[latex]f(-1) = 2(-1)^3 + (-1)^2 - 4(-1) + 1 = 4[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]f(1) = 2(1)^3 + (1)^2 - 4(1) + 1 = 0[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(-\\frac{1}{2}\\right) = 2\\left(-\\frac{1}{2}\\right)^3 + \\left(-\\frac{1}{2}\\right)^2 - 4\\left(-\\frac{1}{2}\\right) + 1 = 3[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(\\frac{1}{2}\\right) = 2\\left(\\frac{1}{2}\\right)^3 + \\left(\\frac{1}{2}\\right)^2 - 4\\left(\\frac{1}{2}\\right) + 1 = -\\frac{1}{2}[\/latex]<\/p>\r\nOf those, [latex]-1, -\\frac{1}{2},[\/latex] and [latex]\\frac{1}{2}[\/latex] are not zeros of [latex]f(x)[\/latex]. [latex]1[\/latex] is the only rational zero of [latex]f(x)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">Use the Rational Zero Theorem to find the rational zeros of [latex]f(x)=x^3\u22125x^2+2x+1[\/latex].[reveal-answer q=\"981143\"]Show Solution[\/reveal-answer] [hidden-answer a=\"981143\"]There are no rational zeros.[\/hidden-answer]<\/section>Watch the following video see more on the rational zero theorem.\r\n\r\n<section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gddbcdee-Iaq7z7reznM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Iaq7z7reznM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gddbcdee-Iaq7z7reznM\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=11328535&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gddbcdee-Iaq7z7reznM&amp;vembed=0&amp;video_id=Iaq7z7reznM&amp;video_target=tpm-plugin-gddbcdee-Iaq7z7reznM\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Finding+All+Zeros+of+a+Polynomial+Function+Using+The+Rational+Zero+Theorem_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding All Zeros of a Polynomial Function Using The Rational Zero Theorem\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fcaecffd-LeZdCSCIb3Q\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/LeZdCSCIb3Q?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fcaecffd-LeZdCSCIb3Q\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12847031&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fcaecffd-LeZdCSCIb3Q&amp;vembed=0&amp;video_id=LeZdCSCIb3Q&amp;video_target=tpm-plugin-fcaecffd-LeZdCSCIb3Q\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Find+the+Zeros+of+a+Polynomial+Function+-+Integer+Zeros_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Find the Zeros of a Polynomial Function - Integer Zeros\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Factor Theorem<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\nThe Factor Theorem is a bridge between zeros and factors of a polynomial. It states that a number [latex] k [\/latex] is a zero of [latex] f(x) [\/latex] if and only if [latex] (x - k) [\/latex] is a factor of [latex] f(x) [\/latex]. This theorem is like a two-way street; knowing a zero lets you find a factor, and knowing a factor lets you find a zero.\r\n\r\nFor instance, to show that [latex] (x + 2) [\/latex] is a factor of [latex] x^3 - 6x^2 - x + 30 [\/latex], you can use synthetic division. If the remainder is zero, then [latex] (x + 2) [\/latex] is indeed a factor, and you can further factorize the quotient to find the remaining zeros of the polynomial.\r\n\r\n<\/div>\r\n<section class=\"textbox example\">Use the Factor Theorem to find the zeros of [latex]f(x)=x^3+4x^2\u22124x\u221216[\/latex] given that [latex](x\u22122)[\/latex] is a factor of the polynomial.[reveal-answer q=\"981336\"]Show Solution[\/reveal-answer] [hidden-answer a=\"981336\"]We can use synthetic division to show that [latex](x -2)[\/latex] is a factor of the polynomial.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c|cccc} 2 &amp; 1 &amp; 4 &amp; -4 &amp; -16 \\\\ &amp; &amp; 2 &amp; 12 &amp; 16 \\\\ \\hline &amp; 1 &amp; 6 &amp; 8 &amp; 0 \\\\ \\end{array}[\/latex]<\/p>\r\nThe remainder is zero, indicating that [latex](x-2)[\/latex] is indeed a factor of the polynomial. Using the Division Algorithm, we can express the polynomial as a product of the divisor and the quotient:\r\n<p style=\"text-align: center;\">[latex] (x-2)(x^2 + 6x + 8) [\/latex]<\/p>\r\nNow, let's further factor the quadratic polynomial [latex]x^2 + 6x + 8[\/latex].\r\n\r\nUsing the middle term factorization method, the factors can be derived as:\r\n<p style=\"text-align: center;\">[latex]x^2 + 4x + 2x + 8[\/latex]<\/p>\r\nwhich gives:\r\n<p style=\"text-align: center;\">[latex]x(x + 4) + 2(x + 4)[\/latex]<\/p>\r\nresulting in:\r\n<p style=\"text-align: center;\">[latex](x + 4)(x + 2)[\/latex]<\/p>\r\nCombining with the previously found factor, we get:\r\n<p style=\"text-align: center;\">[latex](x-2)(x + 4)(x + 2)[\/latex]<\/p>\r\nConcluding from the Factor Theorem, the zeros of [latex]f(x) = x^3 + 4x^2 - 4x - 16[\/latex] are [latex]x = 2[\/latex], [latex]x = -2[\/latex], and [latex]x = -4[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hgfcheha-F0bcRJfdh5Q\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/F0bcRJfdh5Q?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hgfcheha-F0bcRJfdh5Q\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12847032&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hgfcheha-F0bcRJfdh5Q&amp;vembed=0&amp;video_id=F0bcRJfdh5Q&amp;video_target=tpm-plugin-hgfcheha-F0bcRJfdh5Q\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/(New+Version+Available)+Polynomial+Function+-+Complex+Factorization+Theorem_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c(New Version Available) Polynomial Function - Complex Factorization Theorem\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>The Fundamental Theorem of Algebra<\/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\">Every non-constant polynomial function has at least one complex zero.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For a polynomial of degree [latex]n &gt; 0[\/latex], it can be written as: [latex]f(x) = a(x-c_1)(x-c_2)...(x-c_n)[\/latex] where [latex]a[\/latex] is a non-zero real number and [latex]c_1, c_2, ..., c_n[\/latex] are complex numbers.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">A polynomial of degree [latex]n[\/latex] has exactly [latex]n[\/latex] roots, counting multiplicities.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Complex zeros can be real or imaginary.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The Fundamental Theorem of Algebra is crucial for solving polynomial equations.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fcagfegb-d8-LO6FCna0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/d8-LO6FCna0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fcagfegb-d8-LO6FCna0\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12847033&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fcagfegb-d8-LO6FCna0&amp;vembed=0&amp;video_id=d8-LO6FCna0&amp;video_target=tpm-plugin-fcagfegb-d8-LO6FCna0\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Fundamental+theorem+of+algebra+%7C+Polynomial+and+rational+functions+%7C+Algebra+II+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFundamental theorem of algebra | Polynomial and rational functions | Algebra II | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>&nbsp;\r\n<h2>Linear Factorization Theorem and Complex Conjugate Theorem<\/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\">Linear Factorization Theorem:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A polynomial of degree n has exactly n linear factors.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each factor is of the form [latex](x - c)[\/latex], where c is a complex number.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Complex Conjugate Theorem:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For polynomials with real coefficients, complex zeros always occur in conjugate pairs.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]a + bi[\/latex] is a zero, then [latex]a - bi[\/latex] is also a zero.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factoring polynomials:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Real zeros produce factors like [latex](x - r)[\/latex] where r is real.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Complex zeros produce pairs of factors: [latex](x - (a+bi))(x - (a-bi))[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reconstructing polynomials:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Given the zeros and a point on the graph, you can reconstruct the entire polynomial.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find a third degree polynomial with real coefficients that has zeros of 5 and \u20132<em>i<\/em>\u00a0such that [latex]f\\left(1\\right)=10[\/latex].[reveal-answer q=\"704164\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"704164\"][latex]f\\left(x\\right)=-\\frac{1}{2}{x}^{3}+\\frac{5}{2}{x}^{2}-2x+10[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-eecgeggh-4P6MfqEHB90\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/4P6MfqEHB90?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-eecgeggh-4P6MfqEHB90\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780727&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-eecgeggh-4P6MfqEHB90&amp;vembed=0&amp;video_id=4P6MfqEHB90&amp;video_target=tpm-plugin-eecgeggh-4P6MfqEHB90\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Find+a+Polynomial+with+Real+Coefficients+that+has+the+Given+Zeros_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFind a Polynomial with Real Coefficients that has the Given Zeros\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>&nbsp;\r\n<h2>Descartes\u2019 Rule of Signs<\/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\">Descartes' Rule of Signs provides a method to determine the possible numbers of positive and negative real zeros in a polynomial function.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For positive real zeros:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Count the number of sign changes in [latex]f(x)[\/latex] when written in descending order.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The number of positive real zeros is either equal to this count or less than it by an even integer.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For negative real zeros:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Replace [latex]x[\/latex] with [latex]-x[\/latex] in [latex]f(x)[\/latex] to get [latex]f(-x)[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Count the number of sign changes in [latex]f(-x)[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The number of negative real zeros is either equal to this count or less than it by an even integer.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The rule doesn't give the exact number of real zeros, but provides upper bounds and parity information.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Use Descartes\u2019 Rule of Signs to determine the maximum possible number of positive and negative real zeros for [latex]f\\left(x\\right)=2{x}^{4}-10{x}^{3}+11{x}^{2}-15x+12[\/latex].\u00a0Use a graph to verify the number of positive and negative real zeros for the function.[reveal-answer q=\"941865\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"941865\"]There must be 4, 2, or 0 positive real roots and 0 negative real roots. The graph shows that there are 2 positive real zeros and 0 negative real zeros.[\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-badfdgaf-2w1lXuM6pbA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/2w1lXuM6pbA?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-badfdgaf-2w1lXuM6pbA\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780728&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-badfdgaf-2w1lXuM6pbA&amp;vembed=0&amp;video_id=2w1lXuM6pbA&amp;video_target=tpm-plugin-badfdgaf-2w1lXuM6pbA\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/How+to+use+Descartes+rule+of+signs+to+determine+the+number+of+positive+and+negative+zeros_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to use Descartes rule of signs to determine the number of positive and negative zeros\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Evaluate a polynomial using the Remainder Theorem.<\/li>\n<li>Use the Factor Theorem to solve a polynomial equation.<\/li>\n<li>Use the Rational Zero Theorem to find rational zeros.<\/li>\n<li>Find zeros of a polynomial function.<\/li>\n<li>Use the Linear Factorization Theorem to find polynomials with given zeros.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-edhehhea-D_I11k2DfCg\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/D_I11k2DfCg?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-edhehhea-D_I11k2DfCg\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12847030&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-edhehhea-D_I11k2DfCg&amp;vembed=0&amp;video_id=D_I11k2DfCg&amp;video_target=tpm-plugin-edhehhea-D_I11k2DfCg\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+2+-+Find+the+Zeros+of+a+Polynomial+Function+-+Real+Rational+Zeros_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Find the Zeros of a Polynomial Function &#8211; Real Rational Zeros\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Rational Zero Theorem<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>When we&#8217;re navigating the complex sea of polynomial functions, the Rational Zero Theorem is like our compass. It helps us pinpoint potential rational zeros using a simple yet powerful strategy: looking at the factors of the constant term and the leading coefficient.<\/p>\n<p>For instance, consider a polynomial function that has zeros at [latex]\\frac{2}{5}[\/latex] and [latex]\\frac{3}{4}[\/latex]. These aren&#8217;t just numbers; they&#8217;re clues. By setting up equations with these zeros and constructing a quadratic function, we can see a pattern emerge. The numerators of these zeros ([latex]2[\/latex] and [latex]3[\/latex]) are factors of the constant term, while the denominators ([latex]5[\/latex] and [latex]4[\/latex]) are factors of the leading coefficient.<\/p>\n<p><strong>Quick Tips: Applying the Rational Zero Theorem<\/strong><\/p>\n<ol style=\"list-style-type: decimal;\">\n<li><strong>List Out Factors<\/strong>: Start by listing all factors of the constant term and the leading coefficient of your polynomial.<\/li>\n<li><strong>Form Ratios<\/strong>: Create all possible ratios [latex]\\frac{p}{q}[\/latex] where [latex]p[\/latex] is a factor of the constant term and [latex]q[\/latex] is a factor of the leading coefficient.<\/li>\n<li><strong>Test Your Candidates<\/strong>: Evaluate each potential zero by plugging it into the polynomial. If the result is zero, you&#8217;ve found a true zero!<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Use the Rational Zero Theorem to find the rational zeros of [latex]f(x) = 2x^3 + x^2 - 4x + 1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q981243\">Show Solution<\/button> <\/p>\n<div id=\"q981243\" class=\"hidden-answer\" style=\"display: none\">The Rational Zero Theorem tells us that if [latex]\\frac{p}{q}[\/latex] is a zero of [latex]f(x)[\/latex], then [latex]p[\/latex] is a factor of [latex]1[\/latex] and [latex]q[\/latex] is a factor of [latex]2[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c c} \\frac{p}{q} & = \\frac{\\text{factor of constant term}}{\\text{factor of leading coefficient}} \\\\ & = \\frac{\\text{factor of 1}}{\\text{factor of 2}} \\end{array}[\/latex]<\/p>\n<p>The factors of [latex]1[\/latex] are [latex]\\pm1[\/latex] and the factors of [latex]2[\/latex] are [latex]\\pm1[\/latex] and [latex]\\pm2[\/latex]. The possible values for [latex]\\frac{p}{q}[\/latex] are [latex]\\pm1[\/latex] and [latex]\\pm\\frac{1}{2}[\/latex]. These are the possible rational zeros for the function. We can determine which of the possible zeros are actual zeros by substituting these values for [latex]x[\/latex] in [latex]f(x)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f(-1) = 2(-1)^3 + (-1)^2 - 4(-1) + 1 = 4[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]f(1) = 2(1)^3 + (1)^2 - 4(1) + 1 = 0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(-\\frac{1}{2}\\right) = 2\\left(-\\frac{1}{2}\\right)^3 + \\left(-\\frac{1}{2}\\right)^2 - 4\\left(-\\frac{1}{2}\\right) + 1 = 3[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(\\frac{1}{2}\\right) = 2\\left(\\frac{1}{2}\\right)^3 + \\left(\\frac{1}{2}\\right)^2 - 4\\left(\\frac{1}{2}\\right) + 1 = -\\frac{1}{2}[\/latex]<\/p>\n<p>Of those, [latex]-1, -\\frac{1}{2},[\/latex] and [latex]\\frac{1}{2}[\/latex] are not zeros of [latex]f(x)[\/latex]. [latex]1[\/latex] is the only rational zero of [latex]f(x)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Use the Rational Zero Theorem to find the rational zeros of [latex]f(x)=x^3\u22125x^2+2x+1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q981143\">Show Solution<\/button> <\/p>\n<div id=\"q981143\" class=\"hidden-answer\" style=\"display: none\">There are no rational zeros.<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video see more on the rational zero theorem.<\/p>\n<section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gddbcdee-Iaq7z7reznM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Iaq7z7reznM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gddbcdee-Iaq7z7reznM\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=11328535&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gddbcdee-Iaq7z7reznM&amp;vembed=0&amp;video_id=Iaq7z7reznM&amp;video_target=tpm-plugin-gddbcdee-Iaq7z7reznM\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Finding+All+Zeros+of+a+Polynomial+Function+Using+The+Rational+Zero+Theorem_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding All Zeros of a Polynomial Function Using The Rational Zero Theorem\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fcaecffd-LeZdCSCIb3Q\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/LeZdCSCIb3Q?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fcaecffd-LeZdCSCIb3Q\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12847031&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fcaecffd-LeZdCSCIb3Q&amp;vembed=0&amp;video_id=LeZdCSCIb3Q&amp;video_target=tpm-plugin-fcaecffd-LeZdCSCIb3Q\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Find+the+Zeros+of+a+Polynomial+Function+-+Integer+Zeros_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Find the Zeros of a Polynomial Function &#8211; Integer Zeros\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Factor Theorem<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>The Factor Theorem is a bridge between zeros and factors of a polynomial. It states that a number [latex]k[\/latex] is a zero of [latex]f(x)[\/latex] if and only if [latex](x - k)[\/latex] is a factor of [latex]f(x)[\/latex]. This theorem is like a two-way street; knowing a zero lets you find a factor, and knowing a factor lets you find a zero.<\/p>\n<p>For instance, to show that [latex](x + 2)[\/latex] is a factor of [latex]x^3 - 6x^2 - x + 30[\/latex], you can use synthetic division. If the remainder is zero, then [latex](x + 2)[\/latex] is indeed a factor, and you can further factorize the quotient to find the remaining zeros of the polynomial.<\/p>\n<\/div>\n<section class=\"textbox example\">Use the Factor Theorem to find the zeros of [latex]f(x)=x^3+4x^2\u22124x\u221216[\/latex] given that [latex](x\u22122)[\/latex] is a factor of the polynomial.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q981336\">Show Solution<\/button> <\/p>\n<div id=\"q981336\" class=\"hidden-answer\" style=\"display: none\">We can use synthetic division to show that [latex](x -2)[\/latex] is a factor of the polynomial.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c|cccc} 2 & 1 & 4 & -4 & -16 \\\\ & & 2 & 12 & 16 \\\\ \\hline & 1 & 6 & 8 & 0 \\\\ \\end{array}[\/latex]<\/p>\n<p>The remainder is zero, indicating that [latex](x-2)[\/latex] is indeed a factor of the polynomial. Using the Division Algorithm, we can express the polynomial as a product of the divisor and the quotient:<\/p>\n<p style=\"text-align: center;\">[latex](x-2)(x^2 + 6x + 8)[\/latex]<\/p>\n<p>Now, let&#8217;s further factor the quadratic polynomial [latex]x^2 + 6x + 8[\/latex].<\/p>\n<p>Using the middle term factorization method, the factors can be derived as:<\/p>\n<p style=\"text-align: center;\">[latex]x^2 + 4x + 2x + 8[\/latex]<\/p>\n<p>which gives:<\/p>\n<p style=\"text-align: center;\">[latex]x(x + 4) + 2(x + 4)[\/latex]<\/p>\n<p>resulting in:<\/p>\n<p style=\"text-align: center;\">[latex](x + 4)(x + 2)[\/latex]<\/p>\n<p>Combining with the previously found factor, we get:<\/p>\n<p style=\"text-align: center;\">[latex](x-2)(x + 4)(x + 2)[\/latex]<\/p>\n<p>Concluding from the Factor Theorem, the zeros of [latex]f(x) = x^3 + 4x^2 - 4x - 16[\/latex] are [latex]x = 2[\/latex], [latex]x = -2[\/latex], and [latex]x = -4[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hgfcheha-F0bcRJfdh5Q\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/F0bcRJfdh5Q?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hgfcheha-F0bcRJfdh5Q\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12847032&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hgfcheha-F0bcRJfdh5Q&amp;vembed=0&amp;video_id=F0bcRJfdh5Q&amp;video_target=tpm-plugin-hgfcheha-F0bcRJfdh5Q\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/(New+Version+Available)+Polynomial+Function+-+Complex+Factorization+Theorem_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c(New Version Available) Polynomial Function &#8211; Complex Factorization Theorem\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>The Fundamental Theorem of Algebra<\/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\">Every non-constant polynomial function has at least one complex zero.<\/li>\n<li class=\"whitespace-normal break-words\">For a polynomial of degree [latex]n > 0[\/latex], it can be written as: [latex]f(x) = a(x-c_1)(x-c_2)...(x-c_n)[\/latex] where [latex]a[\/latex] is a non-zero real number and [latex]c_1, c_2, ..., c_n[\/latex] are complex numbers.<\/li>\n<li class=\"whitespace-normal break-words\">A polynomial of degree [latex]n[\/latex] has exactly [latex]n[\/latex] roots, counting multiplicities.<\/li>\n<li class=\"whitespace-normal break-words\">Complex zeros can be real or imaginary.<\/li>\n<li class=\"whitespace-normal break-words\">The Fundamental Theorem of Algebra is crucial for solving polynomial equations.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fcagfegb-d8-LO6FCna0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/d8-LO6FCna0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fcagfegb-d8-LO6FCna0\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12847033&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fcagfegb-d8-LO6FCna0&amp;vembed=0&amp;video_id=d8-LO6FCna0&amp;video_target=tpm-plugin-fcagfegb-d8-LO6FCna0\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Fundamental+theorem+of+algebra+%7C+Polynomial+and+rational+functions+%7C+Algebra+II+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFundamental theorem of algebra | Polynomial and rational functions | Algebra II | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>&nbsp;<\/p>\n<h2>Linear Factorization Theorem and Complex Conjugate Theorem<\/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\">Linear Factorization Theorem:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A polynomial of degree n has exactly n linear factors.<\/li>\n<li class=\"whitespace-normal break-words\">Each factor is of the form [latex](x - c)[\/latex], where c is a complex number.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Complex Conjugate Theorem:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For polynomials with real coefficients, complex zeros always occur in conjugate pairs.<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]a + bi[\/latex] is a zero, then [latex]a - bi[\/latex] is also a zero.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Factoring polynomials:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Real zeros produce factors like [latex](x - r)[\/latex] where r is real.<\/li>\n<li class=\"whitespace-normal break-words\">Complex zeros produce pairs of factors: [latex](x - (a+bi))(x - (a-bi))[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reconstructing polynomials:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Given the zeros and a point on the graph, you can reconstruct the entire polynomial.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find a third degree polynomial with real coefficients that has zeros of 5 and \u20132<em>i<\/em>\u00a0such that [latex]f\\left(1\\right)=10[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q704164\">Show Solution<\/button><\/p>\n<div id=\"q704164\" class=\"hidden-answer\" style=\"display: none\">[latex]f\\left(x\\right)=-\\frac{1}{2}{x}^{3}+\\frac{5}{2}{x}^{2}-2x+10[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-eecgeggh-4P6MfqEHB90\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/4P6MfqEHB90?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-eecgeggh-4P6MfqEHB90\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780727&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-eecgeggh-4P6MfqEHB90&amp;vembed=0&amp;video_id=4P6MfqEHB90&amp;video_target=tpm-plugin-eecgeggh-4P6MfqEHB90\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Find+a+Polynomial+with+Real+Coefficients+that+has+the+Given+Zeros_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFind a Polynomial with Real Coefficients that has the Given Zeros\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>&nbsp;<\/p>\n<h2>Descartes\u2019 Rule of Signs<\/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\">Descartes&#8217; Rule of Signs provides a method to determine the possible numbers of positive and negative real zeros in a polynomial function.<\/li>\n<li class=\"whitespace-normal break-words\">For positive real zeros:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Count the number of sign changes in [latex]f(x)[\/latex] when written in descending order.<\/li>\n<li class=\"whitespace-normal break-words\">The number of positive real zeros is either equal to this count or less than it by an even integer.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">For negative real zeros:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Replace [latex]x[\/latex] with [latex]-x[\/latex] in [latex]f(x)[\/latex] to get [latex]f(-x)[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Count the number of sign changes in [latex]f(-x)[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">The number of negative real zeros is either equal to this count or less than it by an even integer.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">The rule doesn&#8217;t give the exact number of real zeros, but provides upper bounds and parity information.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Use Descartes\u2019 Rule of Signs to determine the maximum possible number of positive and negative real zeros for [latex]f\\left(x\\right)=2{x}^{4}-10{x}^{3}+11{x}^{2}-15x+12[\/latex].\u00a0Use a graph to verify the number of positive and negative real zeros for the function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q941865\">Show Solution<\/button><\/p>\n<div id=\"q941865\" class=\"hidden-answer\" style=\"display: none\">There must be 4, 2, or 0 positive real roots and 0 negative real roots. The graph shows that there are 2 positive real zeros and 0 negative real zeros.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-badfdgaf-2w1lXuM6pbA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/2w1lXuM6pbA?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-badfdgaf-2w1lXuM6pbA\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780728&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-badfdgaf-2w1lXuM6pbA&amp;vembed=0&amp;video_id=2w1lXuM6pbA&amp;video_target=tpm-plugin-badfdgaf-2w1lXuM6pbA\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/How+to+use+Descartes+rule+of+signs+to+determine+the+number+of+positive+and+negative+zeros_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to use Descartes rule of signs to determine the number of positive and negative zeros\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":22,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex 2: Find the Zeros of a Polynomial Function - Real Rational Zeros\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/D_I11k2DfCg\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Finding All Zeros of a Polynomial Function Using The Rational Zero Theorem\",\"author\":\"\",\"organization\":\"The Organic Chemistry Tutor\",\"url\":\"https:\/\/youtu.be\/Iaq7z7reznM\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 1: Find the Zeros of a Polynomial Function - Integer 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