{"id":226,"date":"2023-06-21T13:22:46","date_gmt":"2023-06-21T13:22:46","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-zeros-of-polynomial-functions\/"},"modified":"2023-06-21T13:22:46","modified_gmt":"2023-06-21T13:22:46","slug":"summary-zeros-of-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-zeros-of-polynomial-functions\/","title":{"raw":"Summary: Methods for Finding Zeros of Polynomial Functions","rendered":"Summary: Methods for Finding Zeros of Polynomial Functions"},"content":{"raw":"\n\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165135380122\">\n \t<li>To find [latex]f\\left(k\\right)[\/latex], determine the remainder of the polynomial [latex]f\\left(x\\right)[\/latex] when it is divided by [latex]x-k[\/latex].<\/li>\n \t<li><em>k<\/em>&nbsp;is a zero of [latex]f\\left(x\\right)[\/latex] if and only if [latex]\\left(x-k\\right)[\/latex] &nbsp;is a factor of [latex]f\\left(x\\right)[\/latex].<\/li>\n \t<li>Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient.<\/li>\n \t<li>When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.<\/li>\n \t<li>Synthetic division can be used to find the zeros of a polynomial function.<\/li>\n \t<li>According to the Fundamental Theorem of Algebra, every polynomial function has at least one complex zero.<\/li>\n \t<li>Every polynomial function with degree greater than 0 has at least one complex zero.<\/li>\n \t<li>Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form [latex]\\left(x-c\\right)[\/latex] where <em>c<\/em>&nbsp;is a complex number.<\/li>\n \t<li>The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer.<\/li>\n \t<li>The number of negative real zeros of a polynomial function is either the number of sign changes of [latex]f\\left(-x\\right)[\/latex] &nbsp;or less than the number of sign changes by an even integer.<\/li>\n \t<li>Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165133281424\" class=\"definition\">\n \t<dt><strong>Descartes\u2019 Rule of Signs<\/strong><\/dt>\n \t<dd id=\"fs-id1165133281430\">a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of [latex]f\\left(x\\right)[\/latex] and [latex]f\\left(-x\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135459801\" class=\"definition\">\n \t<dt><strong>Factor Theorem<\/strong><\/dt>\n \t<dd id=\"fs-id1165135459806\"><em>k<\/em>&nbsp;is a zero of polynomial function [latex]f\\left(x\\right)[\/latex] if and only if [latex]\\left(x-k\\right)[\/latex] &nbsp;is a factor of [latex]f\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133045332\" class=\"definition\">\n \t<dt><strong>Fundamental Theorem of Algebra<\/strong><\/dt>\n \t<dd id=\"fs-id1165133045337\">a polynomial function with degree greater than 0 has at least one complex zero<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133045341\" class=\"definition\">\n \t<dt><strong>Linear Factorization Theorem<\/strong><\/dt>\n \t<dd id=\"fs-id1165133045347\">allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form [latex]\\left(x-c\\right)[\/latex] where <em>c<\/em>&nbsp;is a complex number<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135456904\" class=\"definition\">\n \t<dt><strong>Rational Zero Theorem<\/strong><\/dt>\n \t<dd id=\"fs-id1165135456910\">the possible rational zeros of a polynomial function have the form [latex]\\frac{p}{q}[\/latex] where <em>p<\/em>&nbsp;is a factor of the constant term and <em>q<\/em>&nbsp;is a factor of the leading coefficient<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137938597\" class=\"definition\">\n \t<dt><strong>Remainder Theorem<\/strong><\/dt>\n \t<dd id=\"fs-id1165137938602\">if a polynomial [latex]f\\left(x\\right)[\/latex] is divided by [latex]x-k[\/latex] , then the remainder is equal to the value [latex]f\\left(k\\right)[\/latex]<\/dd>\n<\/dl>\n\n","rendered":"<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165135380122\">\n<li>To find [latex]f\\left(k\\right)[\/latex], determine the remainder of the polynomial [latex]f\\left(x\\right)[\/latex] when it is divided by [latex]x-k[\/latex].<\/li>\n<li><em>k<\/em>&nbsp;is a zero of [latex]f\\left(x\\right)[\/latex] if and only if [latex]\\left(x-k\\right)[\/latex] &nbsp;is a factor of [latex]f\\left(x\\right)[\/latex].<\/li>\n<li>Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient.<\/li>\n<li>When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.<\/li>\n<li>Synthetic division can be used to find the zeros of a polynomial function.<\/li>\n<li>According to the Fundamental Theorem of Algebra, every polynomial function has at least one complex zero.<\/li>\n<li>Every polynomial function with degree greater than 0 has at least one complex zero.<\/li>\n<li>Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form [latex]\\left(x-c\\right)[\/latex] where <em>c<\/em>&nbsp;is a complex number.<\/li>\n<li>The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer.<\/li>\n<li>The number of negative real zeros of a polynomial function is either the number of sign changes of [latex]f\\left(-x\\right)[\/latex] &nbsp;or less than the number of sign changes by an even integer.<\/li>\n<li>Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165133281424\" class=\"definition\">\n<dt><strong>Descartes\u2019 Rule of Signs<\/strong><\/dt>\n<dd id=\"fs-id1165133281430\">a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of [latex]f\\left(x\\right)[\/latex] and [latex]f\\left(-x\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135459801\" class=\"definition\">\n<dt><strong>Factor Theorem<\/strong><\/dt>\n<dd id=\"fs-id1165135459806\"><em>k<\/em>&nbsp;is a zero of polynomial function [latex]f\\left(x\\right)[\/latex] if and only if [latex]\\left(x-k\\right)[\/latex] &nbsp;is a factor of [latex]f\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133045332\" class=\"definition\">\n<dt><strong>Fundamental Theorem of Algebra<\/strong><\/dt>\n<dd id=\"fs-id1165133045337\">a polynomial function with degree greater than 0 has at least one complex zero<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133045341\" class=\"definition\">\n<dt><strong>Linear Factorization Theorem<\/strong><\/dt>\n<dd id=\"fs-id1165133045347\">allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form [latex]\\left(x-c\\right)[\/latex] where <em>c<\/em>&nbsp;is a complex number<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135456904\" class=\"definition\">\n<dt><strong>Rational Zero Theorem<\/strong><\/dt>\n<dd id=\"fs-id1165135456910\">the possible rational zeros of a polynomial function have the form [latex]\\frac{p}{q}[\/latex] where <em>p<\/em>&nbsp;is a factor of the constant term and <em>q<\/em>&nbsp;is a factor of the leading coefficient<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137938597\" class=\"definition\">\n<dt><strong>Remainder Theorem<\/strong><\/dt>\n<dd id=\"fs-id1165137938602\">if a polynomial [latex]f\\left(x\\right)[\/latex] is divided by [latex]x-k[\/latex] , then the remainder is equal to the value [latex]f\\left(k\\right)[\/latex]<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-226\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":395986,"menu_order":23,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et 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http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"}]","CANDELA_OUTCOMES_GUID":"b5742408-3a4c-4429-b248-4c33fa374f2d","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-226","chapter","type-chapter","status-publish","hentry"],"part":202,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/226","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/users\/395986"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/226\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/202"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/226\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=226"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=226"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=226"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=226"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}