{"id":205,"date":"2023-06-21T13:22:43","date_gmt":"2023-06-21T13:22:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/define-and-identify-polynomial-functions\/"},"modified":"2024-01-08T19:11:28","modified_gmt":"2024-01-08T19:11:28","slug":"define-and-identify-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/define-and-identify-polynomial-functions\/","title":{"raw":"R4.1   Basic Characteristics of Polynomial Functions","rendered":"R4.1   Basic Characteristics of Polynomial Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Determine if a given function is a\u00a0 polynomial function<\/li>\r\n \t<li>Determine the degree and leading coefficient of a polynomial function<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>Recognize Polynomial Functions<\/h3>\r\nWe have introduced polynomials and functions, so now we will combine these ideas to describe polynomial functions.\u00a0Polynomials are algebraic expressions that are created by adding or subtracting monomial terms, such as [latex]-3x^2[\/latex],\u00a0where the exponents are only non-negative integers. Functions are\u00a0a specific type of relation in which each input value has one and only one output value.\u00a0Polynomial functions have all of these characteristics as well as a domain and range, and corresponding graphs. In this section, we will identify and evaluate polynomial functions.\u00a0Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the powers of the variables.\r\n\r\nWhen we introduced polynomials, we presented the following: [latex]4x^3-9x^2+6x[\/latex]. \u00a0We can turn this into a polynomial function by using function notation:\r\n<p style=\"text-align: center;\">[latex]f(x)=4x^3-9x^2+6x[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Polynomial functions are written with the leading term first and all other terms in descending order as a matter of convention. In the first example, we will identify some basic characteristics of polynomial functions.<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p id=\"fs-id1165135262000\">Which of the following are polynomial functions?<\/p>\r\n\r\n<div id=\"eip-id1165134474011\" class=\"equation unnumbered\" style=\"text-align: left;\">[latex]\\begin{array}{ccc}f\\left(x\\right)=5x^7+4\\hfill \\\\ g\\left(x\\right)=-x^2\\left(x-\\dfrac{2}{5}\\right)\\hfill \\\\ h\\left(x\\right)=\\dfrac{1}{2}x^2+\\sqrt{x}+2\\hfill \\end{array}[\/latex]<\/div>\r\n<div><\/div>\r\n[reveal-answer q=\"83362\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"83362\"]\r\n<p id=\"fs-id1165134094645\">The first two functions are examples of polynomial functions because they contain\u00a0powers that are non-negative integers and the coefficients are real numbers. Note that the second function can be written as [latex]g\\left(x\\right)=-x^3+\\dfrac{2}{5}x[\/latex] after applying the distributive property.<\/p>\r\nThe third function is\u00a0not a polynomial function because the variable is under a square root in the middle term, therefore the function contains an exponent that is not a non-negative integer.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you will see additional examples of how to identify a polynomial function using the definition.\r\n\r\nhttps:\/\/youtu.be\/w02qTLrJYiQ\r\n<h2>Determine the Degree and Leading Coefficient\u00a0of a Polynomial Function<\/h2>\r\nJust as we identified the degree of a polynomial, we can identify the degree of a polynomial function. To review: the <strong>degree<\/strong> of the polynomial is the highest power of the variable that occurs in the polynomial; the <strong>leading term<\/strong> is the term containing the highest power of the variable or the term with the highest degree. The <strong>leading coefficient<\/strong> is the coefficient of the leading term.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIdentify the degree, leading term, and leading coefficient of the following polynomial functions.\r\n<p id=\"eip-id1165134242117\" class=\"equation unnumbered\" style=\"text-align: left;\">[latex]\\begin{array}{lll} f\\left(x\\right)=5{x}^{2}+7-4{x}^{3} \\\\ g\\left(x\\right)=9x-{x}^{6}-3{x}^{4}\\\\ h\\left(x\\right)=6\\left(x^2-x\\right)+11\\end{array}[\/latex]<\/p>\r\n<p class=\"equation unnumbered\" style=\"text-align: left;\"><span style=\"font-size: 1rem; text-align: initial;\">[reveal-answer q=\"200839\"]Show Solution[\/reveal-answer][hidden-answer a=\"200839\"]<\/span><\/p>\r\n<p class=\"equation unnumbered\" style=\"text-align: left;\"><span style=\"font-size: 1rem; text-align: initial;\">For the function [latex]f\\left(x\\right)[\/latex], the highest power of\u00a0[latex]x[\/latex]\u00a0is\u00a0[latex]3[\/latex], so the degree is\u00a0[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,\u00a0[latex]\u20134[\/latex].<\/span><\/p>\r\n\r\n<div class=\"equation unnumbered\" style=\"text-align: left;\">\r\n<p id=\"fs-id1165135457771\">For the function [latex]g\\left(x\\right)[\/latex], the highest power of\u00a0[latex]x[\/latex]\u00a0is\u00a0[latex]6[\/latex], so the degree is\u00a0[latex]6[\/latex]. The leading term is the term containing that degree, [latex]-{x}^{6}[\/latex]. The leading coefficient is the coefficient of that term,\u00a0[latex]-1[\/latex].<\/p>\r\n<p id=\"fs-id1165135503949\">For the function [latex]h\\left(x\\right)[\/latex], first rewrite the polynomial using the distributive property to identify the terms. [latex]h\\left(x\\right)=6x^2-6x+11[\/latex]. The highest power of [latex]x[\/latex] is\u00a0[latex]2[\/latex], so the degree is\u00a0[latex]2[\/latex]. The leading term is the term containing that degree, [latex]6{x}^{2}[\/latex]. The leading coefficient is the coefficient of that term,\u00a0[latex]6[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nWatch the next video for more examples of how to identify the degree, leading term and leading coefficient of a polynomial function.\r\n\r\nhttps:\/\/youtu.be\/F_G_w82s0QA\r\n<h2>Summary<\/h2>\r\nPolynomial functions\u00a0contain\u00a0powers that are non-negative integers and the coefficients are real numbers. It is often helpful to know how to\u00a0identify the degree and leading coefficient of a polynomial function. To do this, follow these suggestions:\r\n<ol id=\"fs-id1165135587816\">\r\n \t<li>Find the highest power of <em>x\u00a0<\/em>to determine the degree of the function.<\/li>\r\n \t<li>Identify the term containing the highest power of <em>x\u00a0<\/em>to find the leading term.<\/li>\r\n \t<li>Identify the coefficient of the leading term.<\/li>\r\n<\/ol>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine if a given function is a\u00a0 polynomial function<\/li>\n<li>Determine the degree and leading coefficient of a polynomial function<\/li>\n<\/ul>\n<\/div>\n<h3>Recognize Polynomial Functions<\/h3>\n<p>We have introduced polynomials and functions, so now we will combine these ideas to describe polynomial functions.\u00a0Polynomials are algebraic expressions that are created by adding or subtracting monomial terms, such as [latex]-3x^2[\/latex],\u00a0where the exponents are only non-negative integers. Functions are\u00a0a specific type of relation in which each input value has one and only one output value.\u00a0Polynomial functions have all of these characteristics as well as a domain and range, and corresponding graphs. In this section, we will identify and evaluate polynomial functions.\u00a0Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the powers of the variables.<\/p>\n<p>When we introduced polynomials, we presented the following: [latex]4x^3-9x^2+6x[\/latex]. \u00a0We can turn this into a polynomial function by using function notation:<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=4x^3-9x^2+6x[\/latex]<\/p>\n<p style=\"text-align: left;\">Polynomial functions are written with the leading term first and all other terms in descending order as a matter of convention. In the first example, we will identify some basic characteristics of polynomial functions.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p id=\"fs-id1165135262000\">Which of the following are polynomial functions?<\/p>\n<div id=\"eip-id1165134474011\" class=\"equation unnumbered\" style=\"text-align: left;\">[latex]\\begin{array}{ccc}f\\left(x\\right)=5x^7+4\\hfill \\\\ g\\left(x\\right)=-x^2\\left(x-\\dfrac{2}{5}\\right)\\hfill \\\\ h\\left(x\\right)=\\dfrac{1}{2}x^2+\\sqrt{x}+2\\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q83362\">Show Solution<\/span><\/p>\n<div id=\"q83362\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134094645\">The first two functions are examples of polynomial functions because they contain\u00a0powers that are non-negative integers and the coefficients are real numbers. Note that the second function can be written as [latex]g\\left(x\\right)=-x^3+\\dfrac{2}{5}x[\/latex] after applying the distributive property.<\/p>\n<p>The third function is\u00a0not a polynomial function because the variable is under a square root in the middle term, therefore the function contains an exponent that is not a non-negative integer.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, you will see additional examples of how to identify a polynomial function using the definition.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Determine if a Function is a Polynomial Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/w02qTLrJYiQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Determine the Degree and Leading Coefficient\u00a0of a Polynomial Function<\/h2>\n<p>Just as we identified the degree of a polynomial, we can identify the degree of a polynomial function. To review: the <strong>degree<\/strong> of the polynomial is the highest power of the variable that occurs in the polynomial; the <strong>leading term<\/strong> is the term containing the highest power of the variable or the term with the highest degree. The <strong>leading coefficient<\/strong> is the coefficient of the leading term.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Identify the degree, leading term, and leading coefficient of the following polynomial functions.<\/p>\n<p id=\"eip-id1165134242117\" class=\"equation unnumbered\" style=\"text-align: left;\">[latex]\\begin{array}{lll} f\\left(x\\right)=5{x}^{2}+7-4{x}^{3} \\\\ g\\left(x\\right)=9x-{x}^{6}-3{x}^{4}\\\\ h\\left(x\\right)=6\\left(x^2-x\\right)+11\\end{array}[\/latex]<\/p>\n<p class=\"equation unnumbered\" style=\"text-align: left;\"><span style=\"font-size: 1rem; text-align: initial;\"><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q200839\">Show Solution<\/span><\/p>\n<div id=\"q200839\" class=\"hidden-answer\" style=\"display: none\"><\/span><\/p>\n<p class=\"equation unnumbered\" style=\"text-align: left;\"><span style=\"font-size: 1rem; text-align: initial;\">For the function [latex]f\\left(x\\right)[\/latex], the highest power of\u00a0[latex]x[\/latex]\u00a0is\u00a0[latex]3[\/latex], so the degree is\u00a0[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,\u00a0[latex]\u20134[\/latex].<\/span><\/p>\n<div class=\"equation unnumbered\" style=\"text-align: left;\">\n<p id=\"fs-id1165135457771\">For the function [latex]g\\left(x\\right)[\/latex], the highest power of\u00a0[latex]x[\/latex]\u00a0is\u00a0[latex]6[\/latex], so the degree is\u00a0[latex]6[\/latex]. The leading term is the term containing that degree, [latex]-{x}^{6}[\/latex]. The leading coefficient is the coefficient of that term,\u00a0[latex]-1[\/latex].<\/p>\n<p id=\"fs-id1165135503949\">For the function [latex]h\\left(x\\right)[\/latex], first rewrite the polynomial using the distributive property to identify the terms. [latex]h\\left(x\\right)=6x^2-6x+11[\/latex]. The highest power of [latex]x[\/latex] is\u00a0[latex]2[\/latex], so the degree is\u00a0[latex]2[\/latex]. The leading term is the term containing that degree, [latex]6{x}^{2}[\/latex]. The leading coefficient is the coefficient of that term,\u00a0[latex]6[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the next video for more examples of how to identify the degree, leading term and leading coefficient of a polynomial function.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Degree, Leading Term, and Leading Coefficient of a Polynomial Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/F_G_w82s0QA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>Polynomial functions\u00a0contain\u00a0powers that are non-negative integers and the coefficients are real numbers. It is often helpful to know how to\u00a0identify the degree and leading coefficient of a polynomial function. To do this, follow these suggestions:<\/p>\n<ol id=\"fs-id1165135587816\">\n<li>Find the highest power of <em>x\u00a0<\/em>to determine the degree of the function.<\/li>\n<li>Identify the term containing the highest power of <em>x\u00a0<\/em>to find the leading term.<\/li>\n<li>Identify the coefficient of the leading term.<\/li>\n<\/ol>\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-205\">\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>Determine if a Function is a Polynomial Function. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/w02qTLrJYiQ\">https:\/\/youtu.be\/w02qTLrJYiQ<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Degree, Leading Term, and Leading Coefficient of a Polynomial Function. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/F_G_w82s0QA\">https:\/\/youtu.be\/F_G_w82s0QA<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><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>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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 fro free at : http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Determine if a Function is a Polynomial Function\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen 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