{"id":1398,"date":"2015-11-12T18:35:29","date_gmt":"2015-11-12T18:35:29","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1398"},"modified":"2017-03-31T23:03:21","modified_gmt":"2017-03-31T23:03:21","slug":"key-concepts-glossary-41","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/key-concepts-glossary-41\/","title":{"raw":"Key Concepts &amp; Glossary","rendered":"Key Concepts &amp; Glossary"},"content":{"raw":"<h2 data-type=\"title\">Key Concepts<\/h2>\r\n<ul id=\"fs-id1165135380122\"><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>\r\n\t<li><em>k<\/em>\u00a0is a zero of [latex]f\\left(x\\right)[\/latex]\u00a0if and only if [latex]\\left(x-k\\right)[\/latex]\u00a0is a factor of [latex]f\\left(x\\right)[\/latex].<\/li>\r\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>\r\n\t<li>When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.<\/li>\r\n\t<li>Synthetic division can be used to find the zeros of a polynomial function.<\/li>\r\n\t<li>According to the Fundamental Theorem, every polynomial function has at least one complex zero.<\/li>\r\n\t<li>Every polynomial function with degree greater than 0 has at least one complex zero.<\/li>\r\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>\u00a0is a complex number.<\/li>\r\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>\r\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]\u00a0or less than the number of sign changes by an even integer.<\/li>\r\n\t<li>Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division.<\/li>\r\n<\/ul><h2 data-type=\"glossary-title\">Glossary<\/h2>\r\n<dl id=\"fs-id1165133281424\" class=\"definition\"><dt><strong>Descartes\u2019 Rule of Signs<\/strong><\/dt><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><\/dl><dl id=\"fs-id1165135459801\" class=\"definition\"><dt><strong>Factor Theorem<\/strong><\/dt><dd id=\"fs-id1165135459806\"><em>k<\/em>\u00a0is a zero of polynomial function [latex]f\\left(x\\right)[\/latex] if and only if [latex]\\left(x-k\\right)[\/latex]\u00a0is a factor of [latex]f\\left(x\\right)[\/latex]<\/dd><\/dl><dl id=\"fs-id1165133045332\" class=\"definition\"><dt><strong>Fundamental Theorem of Algebra<\/strong><\/dt><dd id=\"fs-id1165133045337\">a polynomial function with degree greater than 0 has at least one complex zero<\/dd><\/dl><dl id=\"fs-id1165133045341\" class=\"definition\"><dt><strong>Linear Factorization Theorem<\/strong><\/dt><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],\u00a0where <em>c<\/em>\u00a0is a complex number<\/dd><\/dl><dl id=\"fs-id1165135456904\" class=\"definition\"><dt><strong>Rational Zero Theorem<\/strong><\/dt><dd id=\"fs-id1165135456910\">the possible rational zeros of a polynomial function have the form [latex]\\frac{p}{q}[\/latex] where <em>p<\/em>\u00a0is a factor of the constant term and <em>q<\/em>\u00a0is a factor of the leading coefficient.<\/dd><\/dl><dl id=\"fs-id1165137938597\" class=\"definition\"><dt><strong>Remainder Theorem<\/strong><\/dt><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><\/dl>","rendered":"<h2 data-type=\"title\">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>\u00a0is a zero of [latex]f\\left(x\\right)[\/latex]\u00a0if and only if [latex]\\left(x-k\\right)[\/latex]\u00a0is 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, 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>\u00a0is 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]\u00a0or 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 data-type=\"glossary-title\">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>\u00a0is a zero of polynomial function [latex]f\\left(x\\right)[\/latex] if and only if [latex]\\left(x-k\\right)[\/latex]\u00a0is 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],\u00a0where <em>c<\/em>\u00a0is 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>\u00a0is a factor of the constant term and <em>q<\/em>\u00a0is 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-1398\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/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\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/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":276,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1398","chapter","type-chapter","status-publish","hentry"],"part":1376,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1398","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1398\/revisions"}],"predecessor-version":[{"id":2941,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1398\/revisions\/2941"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1376"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1398\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/media?parent=1398"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1398"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1398"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/license?post=1398"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}