{"id":210,"date":"2023-06-21T13:22:44","date_gmt":"2023-06-21T13:22:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/end-behavior-of-polynomial-functions\/"},"modified":"2023-07-04T03:57:23","modified_gmt":"2023-07-04T03:57:23","slug":"end-behavior-of-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/end-behavior-of-polynomial-functions\/","title":{"raw":"\u25aa   End Behavior of Polynomial Functions","rendered":"\u25aa   End Behavior of Polynomial Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify polynomial functions.<\/li>\r\n \t<li>Identify the degree and leading coefficient of polynomial functions.<\/li>\r\n \t<li>Describe the end behavior of a polynomial function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>Identifying Polynomial Functions<\/h3>\r\nAn oil pipeline bursts in the Gulf of Mexico causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius <em>r<\/em>\u00a0of the spill depends on the number of weeks <em>w<\/em>\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 <em>A<\/em>\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<div class=\"textbox\">\r\n<h3>A General Note: Polynomial Functions<\/h3>\r\nLet <em>n<\/em>\u00a0be a non-negative integer. A <strong>polynomial function<\/strong> is a function that can be written in the form\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\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<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying Polynomial Functions<\/h3>\r\nWhich 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<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]48358[\/ohm_question]\r\n\r\n<\/div>\r\n<h3>Defining the Degree and Leading Coefficient\u00a0of a Polynomial Function<\/h3>\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. 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. The <strong>leading term<\/strong> is the term containing the variable with the highest power, also called the term with the highest degree. The <strong>leading coefficient<\/strong> is the coefficient of the leading term.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Terminology of Polynomial Functions<\/h3>\r\n<img class=\"aligncenter\" 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\" \/>\r\n\r\nWe often rearrange polynomials so that the powers on the variable are descending.\r\n\r\nWhen a polynomial is written in this way, we say that it is in general form.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a polynomial function, identify the degree and leading coefficient<\/h3>\r\n<ol>\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>The leading coefficient is the coefficient of the leading term.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying the Degree and Leading Coefficient of a Polynomial Function<\/h3>\r\nIdentify 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 <em>x<\/em>\u00a0is 3, so the degree is 3. 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 <em>t<\/em>\u00a0is 5, so the degree is 5. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, 5.\r\n\r\nFor the function [latex]h\\left(p\\right)[\/latex], the highest power of <em>p<\/em>\u00a0is 3, so the degree is 3. 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<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nIdentify the degree, leading term, and leading coefficient of the polynomial [latex]f\\left(x\\right)=4{x}^{2}-{x}^{6}+2x - 6[\/latex].\r\n\r\n[reveal-answer q=\"435637\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"435637\"]\r\n\r\nThe degree is 6. The leading term is [latex]-{x}^{6}[\/latex]. The leading coefficient is [latex]\u20131[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video for more examples of how to determine the degree, leading term, and leading coefficient of a polynomial.\r\n\r\nhttps:\/\/youtu.be\/F_G_w82s0QA\r\n<h2>Identifying End Behavior of Polynomial Functions<\/h2>\r\nKnowing the leading coefficient and degree of a polynomial function is useful when predicting its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as <em>x<\/em>\u00a0gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the term of highest degree.\r\n<table id=\"Table_03_03_04\" summary=\"..\"><colgroup> <col \/> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center; width: 355px;\">Polynomial Function<\/th>\r\n<th style=\"text-align: center; width: 104px;\">Leading Term<\/th>\r\n<th style=\"text-align: center; width: 89px;\">Graph of Polynomial Function<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 355px;\">[latex]f\\left(x\\right)=5{x}^{4}+2{x}^{3}-x - 4[\/latex]<\/td>\r\n<td style=\"width: 104px;\">[latex]5{x}^{4}[\/latex]<\/td>\r\n<td style=\"width: 89px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194510\/CNX_Precalc_Figure_03_03_0112.jpg\" alt=\"Graph of f(x)=5x^4+2x^3-x-4.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 355px;\">[latex]f\\left(x\\right)=-2{x}^{6}-{x}^{5}+3{x}^{4}+{x}^{3}[\/latex]<\/td>\r\n<td style=\"width: 104px;\">[latex]-2{x}^{6}[\/latex]<\/td>\r\n<td style=\"width: 89px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194512\/CNX_Precalc_Figure_03_03_0122.jpg\" alt=\"Graph of f(x)=-2x^6-x^5+3x^4+x^3.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 355px;\">[latex]f\\left(x\\right)=3{x}^{5}-4{x}^{4}+2{x}^{2}+1[\/latex]<\/td>\r\n<td style=\"width: 104px;\">[latex]3{x}^{5}[\/latex]<\/td>\r\n<td style=\"width: 89px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194514\/CNX_Precalc_Figure_03_03_0132.jpg\" alt=\"Graph of f(x)=3x^5-4x^4+2x^2+1.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 355px;\">[latex]f\\left(x\\right)=-6{x}^{3}+7{x}^{2}+3x+1[\/latex]<\/td>\r\n<td style=\"width: 104px;\">[latex]-6{x}^{3}[\/latex]<\/td>\r\n<td style=\"width: 89px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194516\/CNX_Precalc_Figure_03_03_0142.jpg\" alt=\"Graph of f(x)=-6x^3+7x^2+3x+1.\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying End Behavior and Degree of a Polynomial Function<\/h3>\r\nDescribe the end behavior and determine a possible degree of the polynomial function in the graph below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194520\/CNX_Precalc_Figure_03_03_0152.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"487\" height=\"443\" \/>\r\n\r\n[reveal-answer q=\"626899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"626899\"]\r\n\r\nAs the input values <em>x<\/em>\u00a0get very large, the output values [latex]f\\left(x\\right)[\/latex] increase without bound. As the input values <em>x<\/em>\u00a0get very small, the output values [latex]f\\left(x\\right)[\/latex] decrease without bound. We can describe the end behavior symbolically by writing\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\text{as } x\\to -\\infty , f\\left(x\\right)\\to -\\infty \\\\ \\text{as } x\\to \\infty , f\\left(x\\right)\\to \\infty \\end{array}[\/latex]<\/p>\r\nIn words, we could say that as <em>x<\/em>\u00a0values approach infinity, the function values approach infinity, and as <em>x<\/em>\u00a0values approach negative infinity, the function values approach negative infinity.\r\n\r\nWe can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you'll see more examples that summarize the end behavior of polynomial functions and which components of the function contribute to it.\r\n\r\nhttps:\/\/youtu.be\/y78Dpr9LLN0\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDescribe the end behavior of the polynomial function in the graph below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194522\/CNX_Precalc_Figure_03_03_016n2.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"440\" \/>\r\n\r\n[reveal-answer q=\"304329\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"304329\"]\r\n\r\nAs [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty[\/latex] and as [latex]x\\to -\\infty , f\\left(x\\right)\\to -\\infty [\/latex]. It has the shape of an even degree power function with a negative coefficient.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Tip for success<\/h3>\r\nTo identify the end behavior and degree of a polynomial function, it must be in expanded (general) form. If the function is given to you in factored form, expand it first, then you can identify the leading term.\r\n\r\nPro-tip: You do not have to fully expand the factored form to find the leading term. Note that each of the first terms of the factors multiplied together will give you the leading term.\r\n\r\nEx: given [latex]f(x) = -2x(x+3)(x-3)[\/latex], we know that this will expand as\r\n\r\n[latex]\\begin{align}f(x) &amp;= -2x(x+3)(x-3) \\\\ &amp;=-2x(x^2 - 9) \\\\ &amp;= -2x^3+18x\\end{align}[\/latex]\r\n\r\nBut, multiplying just the first terms together will also reveal the leading term of [latex]-2x^3[\/latex]\r\n\r\n[latex]-2x\\cdot x \\cdot x = -2x^3[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying End Behavior and Degree of a Polynomial Function<\/h3>\r\nGiven the function [latex]f\\left(x\\right)=-3{x}^{2}\\left(x - 1\\right)\\left(x+4\\right)[\/latex], express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.\r\n\r\n[reveal-answer q=\"76137\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"76137\"]\r\n\r\nObtain the general form by expanding the given expression [latex]f\\left(x\\right)[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} f\\left(x\\right)=-3{x}^{2}\\left(x - 1\\right)\\left(x+4\\right)\\\\ f\\left(x\\right)=-3{x}^{2}\\left({x}^{2}+3x - 4\\right)\\\\ f\\left(x\\right)=-3{x}^{4}-9{x}^{3}+12{x}^{2}\\end{array}[\/latex]<\/p>\r\nThe general form is [latex]f\\left(x\\right)=-3{x}^{4}-9{x}^{3}+12{x}^{2}[\/latex].\u00a0The leading term is [latex]-3{x}^{4}[\/latex];\u00a0therefore, the degree of the polynomial is 4. The degree is even (4) and the leading coefficient is negative (\u20133), so the end behavior is\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\text{as } x\\to -\\infty , f\\left(x\\right)\\to -\\infty \\\\ \\text{as } x\\to \\infty , f\\left(x\\right)\\to -\\infty \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven the function [latex]f\\left(x\\right)=0.2\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 5\\right)[\/latex], express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.\r\n\r\n[reveal-answer q=\"657153\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"657153\"]\r\n\r\nThe leading term is [latex]0.2{x}^{3}[\/latex], so it is a degree 3 polynomial. As x approaches positive infinity, [latex]f\\left(x\\right)[\/latex] increases without bound; as x approaches negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound.[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify polynomial functions.<\/li>\n<li>Identify the degree and leading coefficient of polynomial functions.<\/li>\n<li>Describe the end behavior of a polynomial function.<\/li>\n<\/ul>\n<\/div>\n<h3>Identifying Polynomial Functions<\/h3>\n<p>An oil pipeline bursts in the Gulf of Mexico causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius <em>r<\/em>\u00a0of the spill depends on the number of weeks <em>w<\/em>\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 <em>A<\/em>\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<div class=\"textbox\">\n<h3>A General Note: Polynomial Functions<\/h3>\n<p>Let <em>n<\/em>\u00a0be a non-negative integer. A <strong>polynomial function<\/strong> is a function that can be written in the form<\/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>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<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying Polynomial Functions<\/h3>\n<p>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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q906312\">Show Solution<\/span><\/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<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm48358\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=48358&theme=oea&iframe_resize_id=ohm48358&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h3>Defining the Degree and Leading Coefficient\u00a0of a Polynomial Function<\/h3>\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. 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. The <strong>leading term<\/strong> is the term containing the variable with the highest power, also called the term with the highest degree. The <strong>leading coefficient<\/strong> is the coefficient of the leading term.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Terminology of Polynomial Functions<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" 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\" \/><\/p>\n<p>We often rearrange polynomials so that the powers on the variable are descending.<\/p>\n<p>When a polynomial is written in this way, we say that it is in general form.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a polynomial function, identify the degree and leading coefficient<\/h3>\n<ol>\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>The leading coefficient is the coefficient of the leading term.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying the Degree and Leading Coefficient of a Polynomial Function<\/h3>\n<p>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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q632394\">Show Solution<\/span><\/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 <em>x<\/em>\u00a0is 3, so the degree is 3. 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 <em>t<\/em>\u00a0is 5, so the degree is 5. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, 5.<\/p>\n<p>For the function [latex]h\\left(p\\right)[\/latex], the highest power of <em>p<\/em>\u00a0is 3, so the degree is 3. 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<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Identify the degree, leading term, and leading coefficient of the polynomial [latex]f\\left(x\\right)=4{x}^{2}-{x}^{6}+2x - 6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q435637\">Show Solution<\/span><\/p>\n<div id=\"q435637\" class=\"hidden-answer\" style=\"display: none\">\n<p>The degree is 6. The leading term is [latex]-{x}^{6}[\/latex]. The leading coefficient is [latex]\u20131[\/latex].<\/p><\/div>\n<\/div>\n<\/div>\n<p>Watch the following video for more examples of how to determine the degree, leading term, and leading coefficient of a polynomial.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" 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>Identifying End Behavior of Polynomial Functions<\/h2>\n<p>Knowing the leading coefficient and degree of a polynomial function is useful when predicting its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as <em>x<\/em>\u00a0gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the term of highest degree.<\/p>\n<table id=\"Table_03_03_04\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: center; width: 355px;\">Polynomial Function<\/th>\n<th style=\"text-align: center; width: 104px;\">Leading Term<\/th>\n<th style=\"text-align: center; width: 89px;\">Graph of Polynomial Function<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 355px;\">[latex]f\\left(x\\right)=5{x}^{4}+2{x}^{3}-x - 4[\/latex]<\/td>\n<td style=\"width: 104px;\">[latex]5{x}^{4}[\/latex]<\/td>\n<td style=\"width: 89px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194510\/CNX_Precalc_Figure_03_03_0112.jpg\" alt=\"Graph of f(x)=5x^4+2x^3-x-4.\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 355px;\">[latex]f\\left(x\\right)=-2{x}^{6}-{x}^{5}+3{x}^{4}+{x}^{3}[\/latex]<\/td>\n<td style=\"width: 104px;\">[latex]-2{x}^{6}[\/latex]<\/td>\n<td style=\"width: 89px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194512\/CNX_Precalc_Figure_03_03_0122.jpg\" alt=\"Graph of f(x)=-2x^6-x^5+3x^4+x^3.\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 355px;\">[latex]f\\left(x\\right)=3{x}^{5}-4{x}^{4}+2{x}^{2}+1[\/latex]<\/td>\n<td style=\"width: 104px;\">[latex]3{x}^{5}[\/latex]<\/td>\n<td style=\"width: 89px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194514\/CNX_Precalc_Figure_03_03_0132.jpg\" alt=\"Graph of f(x)=3x^5-4x^4+2x^2+1.\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 355px;\">[latex]f\\left(x\\right)=-6{x}^{3}+7{x}^{2}+3x+1[\/latex]<\/td>\n<td style=\"width: 104px;\">[latex]-6{x}^{3}[\/latex]<\/td>\n<td style=\"width: 89px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194516\/CNX_Precalc_Figure_03_03_0142.jpg\" alt=\"Graph of f(x)=-6x^3+7x^2+3x+1.\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying End Behavior and Degree of a Polynomial Function<\/h3>\n<p>Describe the end behavior and determine a possible degree of the polynomial function in the graph below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194520\/CNX_Precalc_Figure_03_03_0152.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"487\" height=\"443\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q626899\">Show Solution<\/span><\/p>\n<div id=\"q626899\" class=\"hidden-answer\" style=\"display: none\">\n<p>As the input values <em>x<\/em>\u00a0get very large, the output values [latex]f\\left(x\\right)[\/latex] increase without bound. As the input values <em>x<\/em>\u00a0get very small, the output values [latex]f\\left(x\\right)[\/latex] decrease without bound. We can describe the end behavior symbolically by writing<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\text{as } x\\to -\\infty , f\\left(x\\right)\\to -\\infty \\\\ \\text{as } x\\to \\infty , f\\left(x\\right)\\to \\infty \\end{array}[\/latex]<\/p>\n<p>In words, we could say that as <em>x<\/em>\u00a0values approach infinity, the function values approach infinity, and as <em>x<\/em>\u00a0values approach negative infinity, the function values approach negative infinity.<\/p>\n<p>We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, you&#8217;ll see more examples that summarize the end behavior of polynomial functions and which components of the function contribute to it.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Summary of End Behavior or Long Run Behavior of Polynomial Functions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/y78Dpr9LLN0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Describe the end behavior of the polynomial function in the graph below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194522\/CNX_Precalc_Figure_03_03_016n2.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"440\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q304329\">Show Solution<\/span><\/p>\n<div id=\"q304329\" class=\"hidden-answer\" style=\"display: none\">\n<p>As [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty[\/latex] and as [latex]x\\to -\\infty , f\\left(x\\right)\\to -\\infty[\/latex]. It has the shape of an even degree power function with a negative coefficient.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Tip for success<\/h3>\n<p>To identify the end behavior and degree of a polynomial function, it must be in expanded (general) form. If the function is given to you in factored form, expand it first, then you can identify the leading term.<\/p>\n<p>Pro-tip: You do not have to fully expand the factored form to find the leading term. Note that each of the first terms of the factors multiplied together will give you the leading term.<\/p>\n<p>Ex: given [latex]f(x) = -2x(x+3)(x-3)[\/latex], we know that this will expand as<\/p>\n<p>[latex]\\begin{align}f(x) &= -2x(x+3)(x-3) \\\\ &=-2x(x^2 - 9) \\\\ &= -2x^3+18x\\end{align}[\/latex]<\/p>\n<p>But, multiplying just the first terms together will also reveal the leading term of [latex]-2x^3[\/latex]<\/p>\n<p>[latex]-2x\\cdot x \\cdot x = -2x^3[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying End Behavior and Degree of a Polynomial Function<\/h3>\n<p>Given the function [latex]f\\left(x\\right)=-3{x}^{2}\\left(x - 1\\right)\\left(x+4\\right)[\/latex], express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q76137\">Show Solution<\/span><\/p>\n<div id=\"q76137\" class=\"hidden-answer\" style=\"display: none\">\n<p>Obtain the general form by expanding the given expression [latex]f\\left(x\\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} f\\left(x\\right)=-3{x}^{2}\\left(x - 1\\right)\\left(x+4\\right)\\\\ f\\left(x\\right)=-3{x}^{2}\\left({x}^{2}+3x - 4\\right)\\\\ f\\left(x\\right)=-3{x}^{4}-9{x}^{3}+12{x}^{2}\\end{array}[\/latex]<\/p>\n<p>The general form is [latex]f\\left(x\\right)=-3{x}^{4}-9{x}^{3}+12{x}^{2}[\/latex].\u00a0The leading term is [latex]-3{x}^{4}[\/latex];\u00a0therefore, the degree of the polynomial is 4. The degree is even (4) and the leading coefficient is negative (\u20133), so the end behavior is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\text{as } x\\to -\\infty , f\\left(x\\right)\\to -\\infty \\\\ \\text{as } x\\to \\infty , f\\left(x\\right)\\to -\\infty \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given the function [latex]f\\left(x\\right)=0.2\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 5\\right)[\/latex], express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q657153\">Show Solution<\/span><\/p>\n<div id=\"q657153\" class=\"hidden-answer\" style=\"display: none\">\n<p>The leading term is [latex]0.2{x}^{3}[\/latex], so it is a degree 3 polynomial. As x approaches positive infinity, [latex]f\\left(x\\right)[\/latex] increases without bound; as x approaches negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound.<\/p><\/div>\n<\/div>\n<\/div>\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-210\">\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>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><li>Question ID 121444. <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>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/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><\/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>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 48358. <strong>Authored by<\/strong>: Wicks, Edward. <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>: IMathAS Community License CC-BY + GPL<\/li><li>Summary of End Behavior or Long Run Behavior of Polynomial Functions . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/y78Dpr9LLN0\">https:\/\/youtu.be\/y78Dpr9LLN0<\/a>. <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>\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":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at 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