{"id":1720,"date":"2016-11-02T20:10:19","date_gmt":"2016-11-02T20:10:19","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=1720"},"modified":"2017-04-14T22:39:33","modified_gmt":"2017-04-14T22:39:33","slug":"end-behavior-of-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/end-behavior-of-polynomial-functions\/","title":{"raw":"Degree and Leading Coefficient","rendered":"Degree and Leading Coefficient"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Describe the end behavior of a polynomial function using it's degree and leading coefficient<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>Identify 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}{c}A\\left(w\\right)=A\\left(r\\left(w\\right)\\right)\\\\ =A\\left(24+8w\\right)\\\\ =\\pi {\\left(24+8w\\right)}^{2}\\end{array}[\/latex]<\/p>\r\nMultiplying gives the formula.\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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"906312\"]\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]\r\ncan be written as [latex]f\\left(x\\right)=6{x}^{4}+4[\/latex].<\/li>\r\n \t<li>[latex]g\\left(x\\right)[\/latex]\r\ncan be written as [latex]g\\left(x\\right)=-{x}^{3}+4x[\/latex].<\/li>\r\n \t<li>[latex]h\\left(x\\right)[\/latex]\r\ncannot 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<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=48358&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>\r\n<h3>Define\u00a0the 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 of power, or in general 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 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\">\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\" data-media-type=\"image\/png\" \/>\r\n\r\nWe often rearrange polynomials so that the powers 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<\/em>\r\nto determine the degree function.<\/li>\r\n \t<li>Identify the term containing the highest power of <em>x<\/em>\r\nto find the leading term.<\/li>\r\n \t<li>Identify 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}{c} 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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"632394\"]\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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"435637\"]The 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\nIn the following video, we show 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 degree of a polynomial function is useful in helping us predict 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 data-align=\"center\" \/> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\" data-align=\"center\">Polynomial Function<\/th>\r\n<th style=\"text-align: center;\" data-align=\"center\">Leading Term<\/th>\r\n<th style=\"text-align: center;\" data-align=\"center\">Graph of Polynomial Function<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td data-valign=\"top\">[latex]f\\left(x\\right)=5{x}^{4}+2{x}^{3}-x - 4[\/latex]<\/td>\r\n<td>[latex]5{x}^{4}[\/latex]<\/td>\r\n<td><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.\" data-media-type=\"image\/jpg\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\">[latex]f\\left(x\\right)=-2{x}^{6}-{x}^{5}+3{x}^{4}+{x}^{3}[\/latex]<\/td>\r\n<td>[latex]-2{x}^{6}[\/latex]<\/td>\r\n<td data-valign=\"top\"><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.\" data-media-type=\"image\/jpg\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\">[latex]f\\left(x\\right)=3{x}^{5}-4{x}^{4}+2{x}^{2}+1[\/latex]<\/td>\r\n<td>[latex]3{x}^{5}[\/latex]<\/td>\r\n<td><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.\" data-media-type=\"image\/jpg\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\">[latex]f\\left(x\\right)=-6{x}^{3}+7{x}^{2}+3x+1[\/latex]<\/td>\r\n<td>[latex]-6{x}^{3}[\/latex]<\/td>\r\n<td><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.\" data-media-type=\"image\/jpg\" \/><\/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\" data-media-type=\"image\/jpg\" \/>\r\n\r\n[reveal-answer q=\"626899\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"626899\"]\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, we show more examples that summarize the end behavior of a 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\" data-media-type=\"image\/jpg\" \/>\r\n[reveal-answer q=\"304329\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"304329\"]\r\n\r\nAs [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty ; as 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 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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"76137\"]\r\nObtain the general form by expanding the given expression for [latex]f\\left(x\\right)[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c} f\\left(x\\right)=-3{x}^{2}\\left(x - 1\\right)\\left(x+4\\right)\\\\ \\hfill =-3{x}^{2}\\left({x}^{2}+3x - 4\\right)\\\\ \\hfill=-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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"657153\"]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.[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Describe the end behavior of a polynomial function using it&#8217;s degree and leading coefficient<\/li>\n<\/ul>\n<\/div>\n<h3>Identify 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}{c}A\\left(w\\right)=A\\left(r\\left(w\\right)\\right)\\\\ =A\\left(24+8w\\right)\\\\ =\\pi {\\left(24+8w\\right)}^{2}\\end{array}[\/latex]<\/p>\n<p>Multiplying gives the formula.<\/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\">Solution<\/span><\/p>\n<div id=\"q906312\" class=\"hidden-answer\" style=\"display: none\">\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.<\/p>\n<ul>\n<li>[latex]f\\left(x\\right)[\/latex]<br \/>\ncan be written as [latex]f\\left(x\\right)=6{x}^{4}+4[\/latex].<\/li>\n<li>[latex]g\\left(x\\right)[\/latex]<br \/>\ncan be written as [latex]g\\left(x\\right)=-{x}^{3}+4x[\/latex].<\/li>\n<li>[latex]h\\left(x\\right)[\/latex]<br \/>\ncannot 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=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=48358&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\n<h3>Define\u00a0the 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 of power, or in general 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 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\">\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\" data-media-type=\"image\/png\" \/><\/p>\n<p>We often rearrange polynomials so that the powers 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<\/em><br \/>\nto determine the degree function.<\/li>\n<li>Identify the term containing the highest power of <em>x<\/em><br \/>\nto find the leading term.<\/li>\n<li>Identify 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}{c} 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\">Solution<\/span><\/p>\n<div id=\"q632394\" class=\"hidden-answer\" style=\"display: none\">\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].<\/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\">Solution<\/span><\/p>\n<div id=\"q435637\" class=\"hidden-answer\" style=\"display: none\">The degree is 6. The leading term is [latex]-{x}^{6}[\/latex]. The leading coefficient is [latex]\u20131[\/latex].<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show 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 degree of a polynomial function is useful in helping us predict 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 data-align=\"center\" \/>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: center;\" data-align=\"center\">Polynomial Function<\/th>\n<th style=\"text-align: center;\" data-align=\"center\">Leading Term<\/th>\n<th style=\"text-align: center;\" data-align=\"center\">Graph of Polynomial Function<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-valign=\"top\">[latex]f\\left(x\\right)=5{x}^{4}+2{x}^{3}-x - 4[\/latex]<\/td>\n<td>[latex]5{x}^{4}[\/latex]<\/td>\n<td><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.\" data-media-type=\"image\/jpg\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">[latex]f\\left(x\\right)=-2{x}^{6}-{x}^{5}+3{x}^{4}+{x}^{3}[\/latex]<\/td>\n<td>[latex]-2{x}^{6}[\/latex]<\/td>\n<td data-valign=\"top\"><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.\" data-media-type=\"image\/jpg\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">[latex]f\\left(x\\right)=3{x}^{5}-4{x}^{4}+2{x}^{2}+1[\/latex]<\/td>\n<td>[latex]3{x}^{5}[\/latex]<\/td>\n<td><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.\" data-media-type=\"image\/jpg\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">[latex]f\\left(x\\right)=-6{x}^{3}+7{x}^{2}+3x+1[\/latex]<\/td>\n<td>[latex]-6{x}^{3}[\/latex]<\/td>\n<td><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.\" data-media-type=\"image\/jpg\" \/><\/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\" data-media-type=\"image\/jpg\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q626899\">Solution<\/span><\/p>\n<div id=\"q626899\" class=\"hidden-answer\" style=\"display: none\">\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<\/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, we show more examples that summarize the end behavior of a 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\" data-media-type=\"image\/jpg\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q304329\">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 ; as 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 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\">Solution<\/span><\/p>\n<div id=\"q76137\" class=\"hidden-answer\" style=\"display: none\">\nObtain the general form by expanding the given expression for [latex]f\\left(x\\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c} f\\left(x\\right)=-3{x}^{2}\\left(x - 1\\right)\\left(x+4\\right)\\\\ \\hfill =-3{x}^{2}\\left({x}^{2}+3x - 4\\right)\\\\ \\hfill=-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\">Solution<\/span><\/p>\n<div id=\"q657153\" class=\"hidden-answer\" style=\"display: none\">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.<\/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-1720\">\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":21,"menu_order":4,"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 http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Question 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