{"id":77,"date":"2025-02-13T22:43:29","date_gmt":"2025-02-13T22:43:29","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/power-functions-and-polynomial-functions\/"},"modified":"2026-01-13T15:39:57","modified_gmt":"2026-01-13T15:39:57","slug":"power-functions-and-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/power-functions-and-polynomial-functions\/","title":{"raw":"Polynomial Functions: Learn It 1","rendered":"Polynomial Functions: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Identify the degree and leading coefficient of polynomial functions.<\/li>\r\n \t<li>Identify end behavior of polynomial functions.<\/li>\r\n \t<li>Identify intercepts of factored polynomial functions.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Polynomial Functions<\/h2>\r\nPolynomial functions allow us to describe curves with multiple peaks and valleys, making them perfect for capturing the intricacies of real-world data.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>polynomial functions<\/h3>\r\nLet [latex]n[\/latex] be a non-negative integer. A <strong>polynomial function<\/strong> is a function that can be written in the form\r\n\r\n&nbsp;\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/p>\r\n&nbsp;\r\n\r\nThis is called the general form of a polynomial function. Each [latex]{a}_{i}[\/latex]\u00a0is a coefficient and can be any real number. Each product [latex]{a}_{i}{x}^{i}[\/latex]\u00a0is a <strong>term of a polynomial function<\/strong>.\r\n\r\n<\/section><section class=\"textbox example\">Which of the following are polynomial functions?\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)=2{x}^{3}\\cdot 3x+4\\hfill \\\\ g\\left(x\\right)=-x\\left({x}^{2}-4\\right)\\hfill \\\\ h\\left(x\\right)=5\\sqrt{x}+2\\hfill \\end{array}[\/latex]<\/p>\r\n[reveal-answer q=\"906312\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"906312\"]\r\n\r\nThe first two functions are examples of polynomial functions because they can be written in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex],\u00a0where the powers are non-negative integers and the coefficients are real numbers.\r\n<ul>\r\n \t<li>[latex]f\\left(x\\right)[\/latex] can be written as [latex]f\\left(x\\right)=6{x}^{4}+4[\/latex].<\/li>\r\n \t<li>[latex]g\\left(x\\right)[\/latex] can be written as [latex]g\\left(x\\right)=-{x}^{3}+4x[\/latex].<\/li>\r\n \t<li>[latex]h\\left(x\\right)[\/latex] cannot be written in this form and is therefore not a polynomial function.<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Degree and Leading Coefficient\u00a0of a Polynomial Function<\/h2>\r\nBecause of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order based on the power on the variable. This is called writing a polynomial in general or standard form.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>terminology of a polynomial function<\/h3>\r\n<ul>\r\n \t<li>The <strong>degree<\/strong> of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form.<\/li>\r\n \t<li>The <strong>leading term<\/strong> is the term containing the variable with the highest power, also called the term with the highest degree.<\/li>\r\n \t<li>The <strong>leading coefficient<\/strong> is the coefficient of the leading term.<\/li>\r\n<\/ul>\r\n<img class=\"wp-image-4961 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02222647\/4.2.2.L1.Diagram-300x149.png\" alt=\"Diagram to show what the components of the leading term in a function are. The leading coefficient is a_n and the degree of the variable is the exponent in x^n. Both the leading coefficient and highest degree variable make up the leading term. So the function looks like f(x)=a_nx^n +\u2026+a_2x^2+a_1x+a_0.\" width=\"378\" height=\"188\" \/>\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a polynomial function, identify the degree and leading coefficient<\/strong>\r\n<ol>\r\n \t<li>Find the highest power of [latex]x[\/latex]to determine the degree of the function.<\/li>\r\n \t<li>Identify the term containing the highest power of [latex]x[\/latex]to find the leading term.<\/li>\r\n \t<li>The leading coefficient is the coefficient of the leading term.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Identify the degree, leading term, and leading coefficient of the following polynomial functions.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} f\\left(x\\right)=3+2{x}^{2}-4{x}^{3} \\\\g\\left(t\\right)=5{t}^{5}-2{t}^{3}+7t\\\\h\\left(p\\right)=6p-{p}^{3}-2\\end{array}[\/latex]<\/p>\r\n[reveal-answer q=\"632394\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"632394\"]\r\n\r\nFor the function [latex]f\\left(x\\right)[\/latex], the highest power of [latex]x[\/latex]\u00a0is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, [latex]\u20134[\/latex].\r\n\r\nFor the function [latex]g\\left(t\\right)[\/latex], the highest power of [latex]t[\/latex]\u00a0is [latex]5[\/latex], so the degree is [latex]5[\/latex]. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, [latex]5[\/latex].\r\n\r\nFor the function [latex]h\\left(p\\right)[\/latex], the highest power of [latex]p[\/latex]\u00a0is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is the term containing that degree, [latex]-{p}^{3}[\/latex]; the leading coefficient is the coefficient of that term, [latex]\u20131[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318816[\/ohm_question]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318818[\/ohm_question]<\/section><\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Identify the degree and leading coefficient of polynomial functions.<\/li>\n<li>Identify end behavior of polynomial functions.<\/li>\n<li>Identify intercepts of factored polynomial functions.<\/li>\n<\/ul>\n<\/section>\n<h2>Polynomial Functions<\/h2>\n<p>Polynomial functions allow us to describe curves with multiple peaks and valleys, making them perfect for capturing the intricacies of real-world data.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>polynomial functions<\/h3>\n<p>Let [latex]n[\/latex] be a non-negative integer. A <strong>polynomial function<\/strong> is a function that can be written in the form<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>This is called the general form of a polynomial function. Each [latex]{a}_{i}[\/latex]\u00a0is a coefficient and can be any real number. Each product [latex]{a}_{i}{x}^{i}[\/latex]\u00a0is a <strong>term of a polynomial function<\/strong>.<\/p>\n<\/section>\n<section class=\"textbox example\">Which of the following are polynomial functions?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)=2{x}^{3}\\cdot 3x+4\\hfill \\\\ g\\left(x\\right)=-x\\left({x}^{2}-4\\right)\\hfill \\\\ h\\left(x\\right)=5\\sqrt{x}+2\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q906312\">Show Solution<\/button><\/p>\n<div id=\"q906312\" class=\"hidden-answer\" style=\"display: none\">\n<p>The first two functions are examples of polynomial functions because they can be written in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex],\u00a0where the powers are non-negative integers and the coefficients are real numbers.<\/p>\n<ul>\n<li>[latex]f\\left(x\\right)[\/latex] can be written as [latex]f\\left(x\\right)=6{x}^{4}+4[\/latex].<\/li>\n<li>[latex]g\\left(x\\right)[\/latex] can be written as [latex]g\\left(x\\right)=-{x}^{3}+4x[\/latex].<\/li>\n<li>[latex]h\\left(x\\right)[\/latex] cannot be written in this form and is therefore not a polynomial function.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<h2>Degree and Leading Coefficient\u00a0of a Polynomial Function<\/h2>\n<p>Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order based on the power on the variable. This is called writing a polynomial in general or standard form.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>terminology of a polynomial function<\/h3>\n<ul>\n<li>The <strong>degree<\/strong> of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form.<\/li>\n<li>The <strong>leading term<\/strong> is the term containing the variable with the highest power, also called the term with the highest degree.<\/li>\n<li>The <strong>leading coefficient<\/strong> is the coefficient of the leading term.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4961 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02222647\/4.2.2.L1.Diagram-300x149.png\" alt=\"Diagram to show what the components of the leading term in a function are. The leading coefficient is a_n and the degree of the variable is the exponent in x^n. Both the leading coefficient and highest degree variable make up the leading term. So the function looks like f(x)=a_nx^n +\u2026+a_2x^2+a_1x+a_0.\" width=\"378\" height=\"188\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02222647\/4.2.2.L1.Diagram-300x149.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02222647\/4.2.2.L1.Diagram-65x32.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02222647\/4.2.2.L1.Diagram-225x112.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02222647\/4.2.2.L1.Diagram-350x174.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02222647\/4.2.2.L1.Diagram.png 427w\" sizes=\"(max-width: 378px) 100vw, 378px\" \/><\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a polynomial function, identify the degree and leading coefficient<\/strong><\/p>\n<ol>\n<li>Find the highest power of [latex]x[\/latex]to determine the degree of the function.<\/li>\n<li>Identify the term containing the highest power of [latex]x[\/latex]to find the leading term.<\/li>\n<li>The leading coefficient is the coefficient of the leading term.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Identify the degree, leading term, and leading coefficient of the following polynomial functions.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} f\\left(x\\right)=3+2{x}^{2}-4{x}^{3} \\\\g\\left(t\\right)=5{t}^{5}-2{t}^{3}+7t\\\\h\\left(p\\right)=6p-{p}^{3}-2\\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q632394\">Show Solution<\/button><\/p>\n<div id=\"q632394\" class=\"hidden-answer\" style=\"display: none\">\n<p>For the function [latex]f\\left(x\\right)[\/latex], the highest power of [latex]x[\/latex]\u00a0is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, [latex]\u20134[\/latex].<\/p>\n<p>For the function [latex]g\\left(t\\right)[\/latex], the highest power of [latex]t[\/latex]\u00a0is [latex]5[\/latex], so the degree is [latex]5[\/latex]. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, [latex]5[\/latex].<\/p>\n<p>For the function [latex]h\\left(p\\right)[\/latex], the highest power of [latex]p[\/latex]\u00a0is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is the term containing that degree, [latex]-{p}^{3}[\/latex]; the leading coefficient is the coefficient of that term, [latex]\u20131[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318816\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318816&theme=lumen&iframe_resize_id=ohm318816&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318818\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318818&theme=lumen&iframe_resize_id=ohm318818&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n","protected":false},"author":6,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":74,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/77"}],"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\/6"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/77\/revisions"}],"predecessor-version":[{"id":5305,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/77\/revisions\/5305"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/74"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/77\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=77"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=77"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=77"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=77"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}