{"id":277,"date":"2019-07-15T22:43:54","date_gmt":"2019-07-15T22:43:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/define-and-identify-polynomial-functions\/"},"modified":"2019-07-15T22:43:54","modified_gmt":"2019-07-15T22:43:54","slug":"define-and-identify-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/define-and-identify-polynomial-functions\/","title":{"raw":"Basic Characteristics of Polynomial Functions","rendered":"Basic Characteristics of Polynomial Functions"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Determine if a given function is a&nbsp; polynomial function<\/li>\n \t<li>Determine the degree and leading coefficient of a polynomial function<\/li>\n<\/ul>\n<\/div>\n<h3>Recognize Polynomial Functions<\/h3>\nWe have introduced polynomials and functions, so now we will combine these ideas to describe polynomial functions.&nbsp;Polynomials are algebraic expressions that are created by adding or subtracting monomial terms, such as [latex]-3x^2[\/latex],&nbsp;where the exponents are only non-negative integers. Functions are&nbsp;a specific type of relation in which each input value has one and only one output value.&nbsp;Polynomial 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.&nbsp;Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the powers of the variables.\n\nWhen we introduced polynomials, we presented the following: [latex]4x^3-9x^2+6x[\/latex]. &nbsp;We can turn this into a polynomial function by using function notation:\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\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p id=\"fs-id1165135262000\">Which of the following are polynomial functions?<\/p>\n\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[reveal-answer q=\"83362\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"83362\"]\n<p id=\"fs-id1165134094645\">The first two functions are examples of polynomial functions because they contain&nbsp;powers 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>\nThe third function is&nbsp;not 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.\n\n[\/hidden-answer]\n\n<\/div>\nIn the following video, you will see additional examples of how to identify a polynomial function using the definition.\n\nhttps:\/\/youtu.be\/w02qTLrJYiQ\n<h2>Determine the Degree and Leading Coefficient&nbsp;of a Polynomial Function<\/h2>\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.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nIdentify the degree, leading term, and leading coefficient of the following polynomial functions.\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\">[reveal-answer q=\"200839\"]Show Solution[\/reveal-answer][hidden-answer a=\"200839\"]<\/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&nbsp;[latex]x[\/latex]&nbsp;is&nbsp;[latex]3[\/latex], so the degree is&nbsp;[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,&nbsp;[latex]\u20134[\/latex].<\/span><\/p>\n\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&nbsp;[latex]x[\/latex]&nbsp;is&nbsp;[latex]6[\/latex], so the degree is&nbsp;[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,&nbsp;[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&nbsp;[latex]2[\/latex], so the degree is&nbsp;[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,&nbsp;[latex]6[\/latex].<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\nWatch the next video for more examples of how to identify the degree, leading term and leading coefficient of a polynomial function.\n\nhttps:\/\/youtu.be\/F_G_w82s0QA\n<h2>Summary<\/h2>\nPolynomial functions&nbsp;contain&nbsp;powers that are non-negative integers and the coefficients are real numbers. It is often helpful to know how to&nbsp;identify the degree and leading coefficient of a polynomial function. To do this, follow these suggestions:\n<ol id=\"fs-id1165135587816\">\n \t<li>Find the highest power of <em>x&nbsp;<\/em>to determine the degree of the function.<\/li>\n \t<li>Identify the term containing the highest power of <em>x&nbsp;<\/em>to find the leading term.<\/li>\n \t<li>Identify the coefficient of the leading term.<\/li>\n<\/ol>\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine if a given function is a&nbsp; 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.&nbsp;Polynomials are algebraic expressions that are created by adding or subtracting monomial terms, such as [latex]-3x^2[\/latex],&nbsp;where the exponents are only non-negative integers. Functions are&nbsp;a specific type of relation in which each input value has one and only one output value.&nbsp;Polynomial 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.&nbsp;Because 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]. &nbsp;We 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&nbsp;powers 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&nbsp;not 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&nbsp;of 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&nbsp;[latex]x[\/latex]&nbsp;is&nbsp;[latex]3[\/latex], so the degree is&nbsp;[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,&nbsp;[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&nbsp;[latex]x[\/latex]&nbsp;is&nbsp;[latex]6[\/latex], so the degree is&nbsp;[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,&nbsp;[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&nbsp;[latex]2[\/latex], so the degree is&nbsp;[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,&nbsp;[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&nbsp;contain&nbsp;powers that are non-negative integers and the coefficients are real numbers. It is often helpful to know how to&nbsp;identify 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&nbsp;<\/em>to determine the degree of the function.<\/li>\n<li>Identify the term containing the highest power of <em>x&nbsp;<\/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-277\">\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":17533,"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 Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/w02qTLrJYiQ\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Degree, Leading Term, and Leading Coefficient of a Polynomial Function\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/F_G_w82s0QA\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay, et al\",\"organization\":\"Open Stax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download fro free at : http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-277","chapter","type-chapter","status-publish","hentry"],"part":274,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/277","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/277\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/parts\/274"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/277\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/media?parent=277"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=277"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/contributor?post=277"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/license?post=277"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}