{"id":788,"date":"2024-04-29T19:20:05","date_gmt":"2024-04-29T19:20:05","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=788"},"modified":"2025-08-21T23:04:33","modified_gmt":"2025-08-21T23:04:33","slug":"polynomial-basics-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/polynomial-basics-learn-it-1\/","title":{"raw":"Polynomial Basics: Learn It 1","rendered":"Polynomial Basics: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Recognize polynomial functions, noting their degree and leading coefficient<\/li>\r\n \t<li>Add, subtract, and multiply polynomials using different methods, including the FOIL method for two-term polynomials<\/li>\r\n \t<li>Work with polynomials that have more than one variable, understanding how to combine and simplify them<\/li>\r\n<\/ul>\r\n<\/section><section>\r\n<div class=\"page\" title=\"Page 70\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n<h2>Identifying Polynomial Functions<\/h2>\r\nAn oil pipeline bursts in the Gulf of Mexico causing an oil slick in a roughly circular shape. The slick is currently [latex]24[\/latex] miles in radius, but that radius is increasing by [latex]8[\/latex] miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius [latex]r[\/latex] of the spill depends on the number of weeks [latex]w[\/latex]\u00a0that have passed. This relationship is linear.\r\n<p style=\"text-align: center;\">[latex]r\\left(w\\right)=24+8w[\/latex]<\/p>\r\nWe can combine this with the formula for the area [latex]A[\/latex]\u00a0of a circle.\r\n<p style=\"text-align: center;\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/p>\r\nComposing these functions gives a formula for the area in terms of weeks.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}A\\left(w\\right)=A\\left(r\\left(w\\right)\\right)\\\\ A\\left(w\\right)=A\\left(24+8w\\right)\\\\ A\\left(w\\right)=\\pi {\\left(24+8w\\right)}^{2}\\end{array}[\/latex]<\/p>\r\nMultiplying gives the formula below.\r\n<p style=\"text-align: center;\">[latex]A\\left(w\\right)=576\\pi +384\\pi w+64\\pi {w}^{2}[\/latex]<\/p>\r\nThis formula is an example of a <strong>polynomial function<\/strong>. A polynomial function consists of either zero or the sum of a finite number of non-zero\u00a0terms, each of which is a product of a number, called the\u00a0coefficient\u00a0of the term, and a variable raised to a non-negative integer power.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>polynomial functions<\/h3>\r\nLet [latex]n[\/latex]\u00a0 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 <strong>general form of a polynomial function<\/strong>.\r\n\r\n&nbsp;\r\n\r\nEach [latex]{a}_{i}[\/latex]\u00a0is a coefficient and can be any real number.\r\n\r\n&nbsp;\r\n\r\nEach product [latex]{a}_{i}{x}^{i}[\/latex]\u00a0is a <strong>term of a polynomial function<\/strong>.\r\n\r\n<\/section>A polynomial can have different names based on the number of terms it contains:\r\n<ul>\r\n \t<li>A polynomial containing only one term, such as [latex]5x^4[\/latex], is called a <strong>monomial<\/strong>.<\/li>\r\n \t<li>A polynomial containing two terms, such as [latex]2x-9[\/latex], is called a <strong>binomial<\/strong>.<\/li>\r\n \t<li>A polynomial containing three terms, such as [latex]3x^2 -8x+5[\/latex], is called a <strong>trinomial<\/strong>.<\/li>\r\n<\/ul>\r\n<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><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]13810[\/ohm2_question]<\/section>\r\n<h2>Defining the Degree and Leading Coefficient of 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 polynomial functions<\/h3>\r\n<center>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194507\/CNX_Precalc_Figure_03_03_010n2.jpg\" 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=\"487\" height=\"147\" \/> Diagram to show what the components of the leading term in a function are[\/caption]\r\n\r\n<\/center>&nbsp;\r\n\r\nThe <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.\r\n\r\n&nbsp;\r\n\r\nThe <strong>leading term<\/strong> is the term containing the variable with the highest power, also called the term with the highest degree.\r\n\r\n&nbsp;\r\n\r\nThe <strong>leading coefficient<\/strong> is the coefficient of the leading term.\r\n\r\n<\/section><section class=\"textbox questionHelp\"><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 <strong>degree of the function<\/strong>.<\/li>\r\n \t<li>Identify the term containing the highest power of [latex]x[\/latex]<em>\u00a0<\/em>to find the <strong>leading term<\/strong>.<\/li>\r\n \t<li>The <strong>leading coefficient<\/strong> 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] is [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\">[ohm2_question height=\"400\" hide_question_numbers=1]13816[\/ohm2_question]<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Recognize polynomial functions, noting their degree and leading coefficient<\/li>\n<li>Add, subtract, and multiply polynomials using different methods, including the FOIL method for two-term polynomials<\/li>\n<li>Work with polynomials that have more than one variable, understanding how to combine and simplify them<\/li>\n<\/ul>\n<\/section>\n<section>\n<div class=\"page\" title=\"Page 70\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<h2>Identifying Polynomial Functions<\/h2>\n<p>An oil pipeline bursts in the Gulf of Mexico causing an oil slick in a roughly circular shape. The slick is currently [latex]24[\/latex] miles in radius, but that radius is increasing by [latex]8[\/latex] miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius [latex]r[\/latex] of the spill depends on the number of weeks [latex]w[\/latex]\u00a0that have passed. This relationship is linear.<\/p>\n<p style=\"text-align: center;\">[latex]r\\left(w\\right)=24+8w[\/latex]<\/p>\n<p>We can combine this with the formula for the area [latex]A[\/latex]\u00a0of a circle.<\/p>\n<p style=\"text-align: center;\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/p>\n<p>Composing these functions gives a formula for the area in terms of weeks.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}A\\left(w\\right)=A\\left(r\\left(w\\right)\\right)\\\\ A\\left(w\\right)=A\\left(24+8w\\right)\\\\ A\\left(w\\right)=\\pi {\\left(24+8w\\right)}^{2}\\end{array}[\/latex]<\/p>\n<p>Multiplying gives the formula below.<\/p>\n<p style=\"text-align: center;\">[latex]A\\left(w\\right)=576\\pi +384\\pi w+64\\pi {w}^{2}[\/latex]<\/p>\n<p>This formula is an example of a <strong>polynomial function<\/strong>. A polynomial function consists of either zero or the sum of a finite number of non-zero\u00a0terms, each of which is a product of a number, called the\u00a0coefficient\u00a0of the term, and a variable raised to a non-negative integer power.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>polynomial functions<\/h3>\n<p>Let [latex]n[\/latex]\u00a0 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 <strong>general form of a polynomial function<\/strong>.<\/p>\n<p>&nbsp;<\/p>\n<p>Each [latex]{a}_{i}[\/latex]\u00a0is a coefficient and can be any real number.<\/p>\n<p>&nbsp;<\/p>\n<p>Each product [latex]{a}_{i}{x}^{i}[\/latex]\u00a0is a <strong>term of a polynomial function<\/strong>.<\/p>\n<\/section>\n<p>A polynomial can have different names based on the number of terms it contains:<\/p>\n<ul>\n<li>A polynomial containing only one term, such as [latex]5x^4[\/latex], is called a <strong>monomial<\/strong>.<\/li>\n<li>A polynomial containing two terms, such as [latex]2x-9[\/latex], is called a <strong>binomial<\/strong>.<\/li>\n<li>A polynomial containing three terms, such as [latex]3x^2 -8x+5[\/latex], is called a <strong>trinomial<\/strong>.<\/li>\n<\/ul>\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<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm13810\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13810&theme=lumen&iframe_resize_id=ohm13810&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Defining the Degree and Leading Coefficient of 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 polynomial functions<\/h3>\n<div style=\"text-align: center;\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194507\/CNX_Precalc_Figure_03_03_010n2.jpg\" 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=\"487\" height=\"147\" \/><figcaption class=\"wp-caption-text\">Diagram to show what the components of the leading term in a function are<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>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.<\/p>\n<p>&nbsp;<\/p>\n<p>The <strong>leading term<\/strong> is the term containing the variable with the highest power, also called the term with the highest degree.<\/p>\n<p>&nbsp;<\/p>\n<p>The <strong>leading coefficient<\/strong> is the coefficient of the leading term.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\"><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 <strong>degree of the function<\/strong>.<\/li>\n<li>Identify the term containing the highest power of [latex]x[\/latex]<em>\u00a0<\/em>to find the <strong>leading term<\/strong>.<\/li>\n<li>The <strong>leading coefficient<\/strong> 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] is [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=\"ohm13816\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13816&theme=lumen&iframe_resize_id=ohm13816&source=tnh\" width=\"100%\" height=\"400\"><\/iframe><\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":12,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":55,"module-header":"learn_it","content_attributions":[],"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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/788"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":23,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/788\/revisions"}],"predecessor-version":[{"id":7963,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/788\/revisions\/7963"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/55"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/788\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=788"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=788"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=788"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=788"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}