{"id":16686,"date":"2020-04-07T21:19:56","date_gmt":"2020-04-07T21:19:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/?post_type=chapter&#038;p=16686"},"modified":"2021-08-23T06:48:19","modified_gmt":"2021-08-23T06:48:19","slug":"polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/polynomial-functions\/","title":{"raw":"Section 4.1 Polynomial Functions","rendered":"Section 4.1 Polynomial Functions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify polynomial functions<\/li>\r\n \t<li>Identify the degree, leading coefficient, and multiplicity of polynomial functions<\/li>\r\n \t<li>Identify end behavior of polynomial functions.<\/li>\r\n \t<li>Writing a formula for a polynomial function from the graph<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>\u00a0Identify polynomial functions<\/h2>\r\n<p id=\"fs-id1165135689465\">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>\r\n\r\n<div id=\"eip-719\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]r\\left(w\\right)=24+8w[\/latex]<\/div>\r\n<p id=\"fs-id1165133432974\">We can combine this with the formula for the area <em>A<\/em>\u00a0of a circle.<\/p>\r\n\r\n<div id=\"eip-731\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/div>\r\n<p id=\"fs-id1165137704887\">Composing these functions gives a formula for the area in terms of weeks.<\/p>\r\n\r\n<div id=\"eip-645\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}A\\left(w\\right)&amp;=A\\left(r\\left(w\\right)\\right)\\\\ &amp;=A\\left(24+8w\\right)\\\\ &amp;=\\pi {\\left(24+8w\\right)}^{2}\\end{align}[\/latex]<\/div>\r\n<p id=\"fs-id1165137835475\">Multiplying gives the formula.<\/p>\r\n\r\n<div id=\"eip-290\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A\\left(w\\right)=576\\pi +384\\pi w+64\\pi {w}^{2}[\/latex]<\/div>\r\n<p id=\"fs-id1165135205726\">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>\r\n\r\n<div id=\"fs-id1165137715427\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Polynomial Functions<\/h3>\r\n<p id=\"fs-id1165137823247\">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>\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<p id=\"eip-id1165137832690\">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>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137715427\" class=\"note textbox\">\r\n<h3 class=\"title\">Characteristics of Polynomial Functions<\/h3>\r\n<ul>\r\n \t<li>A polynomial function consists of either zero or the sum of a finite number of non-zero\u00a0terms<\/li>\r\n \t<li>A polynomial must have whole number exponents [latex](0,1,2,3,...)[\/latex]<\/li>\r\n \t<li>A polynomial cannot have negative, fractional, or decimal exponents<\/li>\r\n \t<li>A polynomial must be a smooth continuous curve with no breaks or cusps<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"Example_03_03_04\" class=\"example\">\r\n<div id=\"fs-id1165137817691\" class=\"exercise\">\r\n<div id=\"fs-id1165137817693\" class=\"problem textbox shaded\">\r\n<h3>Example 1: Identifying Polynomial Functions<\/h3>\r\n<p id=\"fs-id1165135262000\">Which of the following are polynomial functions?<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}f\\left(x\\right)=2{x}^{3}\\cdot 3x+4 \\\\ g\\left(x\\right)=-x\\left({x}^{2}-4\\right) \\\\ h\\left(x\\right)=5\\sqrt{x}+2 \\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"824812\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"824812\"]\r\n<p id=\"fs-id1165134094645\">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>\r\n\r\n<ul id=\"fs-id1165137864157\">\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.\u00a0 In addition, the exponent is [latex]\\frac{1}{2}[\/latex], which means it is not a polynomial.<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h2>\u00a0Identify the degree and leading coefficient of polynomial functions<\/h2>\r\n<p id=\"fs-id1165137831216\">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>\r\n\r\n<div id=\"fs-id1165135193124\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Terminology of Polynomial Functions<\/h3>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010717\/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\" \/> <b>Figure 1<\/b>[\/caption]\r\n<p id=\"fs-id1165137921667\">We often rearrange polynomials so that the powers are descending.<span id=\"fs-id1165137406148\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165137482568\">When a polynomial is written in this way, we say that it is in general form.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134031372\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137803898\">How To: Given a polynomial function, identify the degree and leading coefficient.<\/h3>\r\n<ol id=\"fs-id1165135587816\">\r\n \t<li>Find the highest power of <em>x\u00a0<\/em>to determine the degree function.<\/li>\r\n \t<li>Identify the term containing the highest power of <em>x\u00a0<\/em>to find the leading term.<\/li>\r\n \t<li>Identify the coefficient of the leading term.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_03_05\" class=\"example\">\r\n<div id=\"fs-id1165137401820\" class=\"exercise\">\r\n<div id=\"fs-id1165137862379\" class=\"problem textbox shaded\">\r\n<h3>Example 2: Identifying the Degree and Leading Coefficient of a Polynomial Function<\/h3>\r\n<p id=\"fs-id1165137435372\">Identify the degree, leading term, and leading coefficient of the following polynomial functions.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered} 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{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"951580\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"951580\"]\r\n<p id=\"fs-id1165137722510\">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, \u20134.<\/p>\r\n<p id=\"fs-id1165135457771\">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>\r\n<p id=\"fs-id1165135503949\">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, \u20131.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137424484\">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>\r\n[reveal-answer q=\"148647\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"148647\"]\r\n\r\nThe degree is 6. The leading term is [latex]-{x}^{6}[\/latex]. The leading coefficient is \u20131.\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\n[ohm_question hide_question_numbers=1]34293[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Identifying Local Behavior of Polynomial Functions<\/h2>\r\n<p id=\"fs-id1165134054039\">A <strong>turning point <\/strong>is a point at which the function values change from increasing to decreasing or decreasing to increasing.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010719\/CNX_Precalc_Figure_03_03_0172.jpg\" alt=\"\" width=\"731\" height=\"629\" \/> <b>Figure 2<\/b>[\/caption]\r\n<h2>Understand the relationship between degree and turning points<\/h2>\r\n<p id=\"fs-id1165135416524\">\u00a0Look at the graph of the polynomial function [latex]f\\left(x\\right)={x}^{4}-{x}^{3}-4{x}^{2}+4x[\/latex] in Figure 3. The graph has three turning points.<span id=\"fs-id1165134155116\">\r\n<\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010733\/CNX_Precalc_Figure_03_04_0152.jpg\" alt=\"Graph of an odd-degree polynomial with a negative leading coefficient. Note that as x goes to positive infinity, f(x) goes to negative infinity, and as x goes to negative infinity, f(x) goes to positive infinity.\" width=\"487\" height=\"327\" \/> <b>Figure 3<\/b>[\/caption]\r\n<p id=\"fs-id1165137784439\">This function <em>f<\/em>\u00a0is a 4<sup>th<\/sup> degree polynomial function and has 3 turning points. The maximum number of turning points of a polynomial function is always one less than the degree of the function.<\/p>\r\n\r\n<div id=\"fs-id1165135502799\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Interpreting Turning Points<\/h3>\r\n<p id=\"fs-id1165135469050\">A <strong>turning point<\/strong> is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising).<\/p>\r\n<p id=\"fs-id1165135469055\">A polynomial of degree <em>n<\/em>\u00a0will have at most <em>n<\/em> \u2013 1\u00a0turning points.<\/p>\r\n\r\n<\/div>\r\n<div id=\"Example_03_04_07\" class=\"example\">\r\n<div id=\"fs-id1165134374690\" class=\"exercise\">\r\n<div id=\"fs-id1165134060420\" class=\"problem textbox shaded\">\r\n<h3>Example 3: Finding the Maximum Number of Turning Points Using the Degree of a Polynomial Function<\/h3>\r\n<p id=\"fs-id1165134060425\">Find the maximum number of turning points of each polynomial function.<\/p>\r\n\r\n<ol id=\"fs-id1165134060428\">\r\n \t<li>[latex]f\\left(x\\right)=-{x}^{3}+4{x}^{5}-3{x}^{2}++1[\/latex]<\/li>\r\n \t<li>[latex]f\\left(x\\right)=-{\\left(x - 1\\right)}^{2}\\left(1+2{x}^{2}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"157524\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"157524\"]\r\n<ol id=\"fs-id1165137784430\">\r\n \t<li>[latex]f\\left(x\\right)=-x{}^{3}+4{x}^{5}-3{x}^{2}++1[\/latex]\r\n<p id=\"fs-id1165135335895\">First, rewrite the polynomial function in descending order: [latex]f\\left(x\\right)=4{x}^{5}-{x}^{3}-3{x}^{2}++1[\/latex]<\/p>\r\n<p id=\"fs-id1165135453844\">Identify the degree of the polynomial function. This polynomial function is of degree 5.<\/p>\r\n<p id=\"fs-id1165135341233\">The maximum number of turning points is 5 \u2013 1 = 4.<\/p>\r\n<\/li>\r\n \t<li>[latex]f\\left(x\\right)=-{\\left(x - 1\\right)}^{2}\\left(1+2{x}^{2}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2019\/04\/01021335\/CNX_Precalc_Figure_03_04_0162.jpg\"><img class=\"aligncenter wp-image-15117 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2019\/04\/01021335\/CNX_Precalc_Figure_03_04_0162.jpg\" alt=\"Graphic of f(x) showing to multiply the first term of (x-1)^2 and 2x^2 to determine the leading term.\" width=\"487\" height=\"67\" \/><\/a>\r\n<p style=\"text-align: center;\">[latex]a_{n}=-\\left(x^2\\right)\\left(2x^2\\right)=-2x^4[\/latex]<\/p>\r\n<p id=\"fs-id1165133104532\">First, identify the leading term of the polynomial function if the function were expanded.<span id=\"fs-id1165134130071\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165135551181\">Then, identify the degree of the polynomial function. This polynomial function is of degree 4.<\/p>\r\n<p id=\"fs-id1165135551185\">The maximum number of turning points is 4 \u2013 1 = 3.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135188274\">Without graphing the function, determine the maximum number of\u00a0turning points for [latex]f\\left(x\\right)=108 - 13{x}^{9}-8{x}^{4}+14{x}^{12}+2{x}^{3}[\/latex]<\/p>\r\n[reveal-answer q=\"515707\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"515707\"]\r\n\r\nThere are at most\u00a011 turning points.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Comparing Smooth and Continuous Graphs<\/span>\r\n\r\n<section id=\"fs-id1165134080932\">\r\n<p id=\"fs-id1165137692509\">The degree of a polynomial function helps us to determine\u00a0the number of turning points. A polynomial function of\u00a0<em>n<\/em>th degree is the product of <em>n<\/em>\u00a0factors, so it will have at most <em>n<\/em>\u00a0roots or zeros, or <em>x<\/em>-intercepts. The graph of the polynomial function of degree <em>n<\/em>\u00a0must have at most <em>n<\/em> \u2013 1\u00a0turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.<\/p>\r\n<p id=\"fs-id1165137657937\">A <strong>continuous function<\/strong> has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A <strong>smooth curve<\/strong> is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.<\/p>\r\n\r\n<div id=\"Example_03_04_01\" class=\"example\">\r\n<div id=\"fs-id1165137643218\" class=\"exercise\">\r\n<div id=\"fs-id1165133360328\" class=\"problem textbox shaded\">\r\n<h3>Example 4: Recognizing Polynomial Functions<\/h3>\r\nWhich of the graphs in Figure 4\u00a0represents a degree 3 polynomial function?\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010727\/CNX_Precalc_Figure_03_04_0022.jpg\" alt=\"Two graphs in which one has a polynomial function and the other has a function closely resembling a polynomial but is not.\" width=\"731\" height=\"766\" \/> <b>Figure 4<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"898519\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"898519\"]\r\n<p id=\"fs-id1165134188794\">The graphs of <em>g<\/em>\u00a0and <em>k\u00a0<\/em>are graphs of functions that are not polynomials. The graph of function <em>g<\/em>\u00a0has a sharp corner. The graph of function <em>k<\/em>\u00a0is not continuous.<\/p>\r\nThe graph of f is a polynomial, however it has 4 turning points, which is too many for a degree 3 polynomial.\r\n\r\nThe graph if h could be a degree 3 polynomial.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165137735781\">\r\n<div id=\"fs-id1165137766902\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137645233\">The Factor Theorem<\/h3>\r\nFor the polynomial [latex]P(x)[\/latex],\r\n<ol id=\"fs-id1165137571388\">\r\n \t<li>If [latex]r[\/latex] is a zero of [latex]P(x)[\/latex] then [latex]x-r[\/latex] will be a factor of [latex]P(x)[\/latex].<\/li>\r\n \t<li>If [latex]x-r[\/latex] is a factor of [latex]P(x)[\/latex] then [latex]r[\/latex] will be a zero of [latex]P(x)[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_03_08\" class=\"example\">\r\n<div id=\"fs-id1165137435581\" class=\"exercise\">\r\n<div id=\"fs-id1165137803210\" class=\"problem textbox shaded\">\r\n<h3>Example 5: Form a polynomial from given zeros<\/h3>\r\nGiven [latex]f(x)[\/latex] has zeros of [latex]-5,0,5[\/latex], form a degree three polynomial with integer coefficients.\r\n[reveal-answer q=\"994840\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"994840\"]\r\nTo form the polynomial, we will use part 2 of the factor theorem. We will use the formula [latex]f(x)=\\left(x-r_{1}\\right)\\left(x-r_{2}\\right)\\left(x-r_{3}\\right)... \\\\ f(x)=\\left(x-(-5)\\right)\\left(x-0\\right)\\left(x-5\\right) \\\\ f(x)=x(x+5)(x-5) \\\\ f(x) = x^{3}-25x[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_03_08\" class=\"example\">\r\n<div id=\"fs-id1165137435581\" class=\"exercise\">\r\n<div id=\"fs-id1165137803210\" class=\"problem textbox shaded\">\r\n<h3>Example 6: Form a polynomial from given zeros<\/h3>\r\nGiven [latex]f(x)[\/latex] has zeros of [latex]-3,2[\/latex], form a degree three polynomial with integer coefficients.\r\n[reveal-answer q=\"994850\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"994850\"]\r\nTo form the polynomial, we will use part 2 of the factor theorem. We will use the formula: [latex]f(x)=\\left(x-r_{1}\\right)\\left(x-r_{2}\\right)\\left(x-r_{3}\\right)... \\\\ f(x)=\\left(x-(-3)\\right)\\left(x-2\\right) \\\\ f(x)=(x+3)(x-2)[\/latex]\r\n\r\nHowever, [latex]f(x)=(x+3)(x-2)[\/latex] is not a degree 3 polynomial. To make it a degree three, [latex]f(x)[\/latex] can be written two different ways:\r\n[latex]f(x)=(x+3)^{2}(x-2)\\text{ or} \\\\ f(x)=(x+3)(x-2)^{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn the previous example, writing a degree polynomial with zeros of [latex]-3,2[\/latex] resulted in two answers: [latex]f(x)=(x+3)^{2}(x-2)[\/latex] or [latex]f(x)=(x+3)(x-2)^{2}[\/latex]. The exponents on each of the factors are called <strong>multiplicities<\/strong>.\r\n\r\nMultiplicities are useful because they indicate whether a graph touches or crosses the x-axis.\r\n<h2>Identify zeros and their multiplicities<\/h2>\r\n<p id=\"fs-id1165135581073\">Graphs behave differently at various <em>x<\/em>-intercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and bounce off.<\/p>\r\n<p id=\"fs-id1165133092720\">Suppose, for example, we graph the function<\/p>\r\n\r\n<div id=\"eip-840\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=\\left(x+3\\right){\\left(x - 2\\right)}^{2}{\\left(x+1\\right)}^{3}[\/latex].<\/div>\r\nNotice in Figure 5\u00a0that the behavior of the function at each of the <em>x<\/em>-intercepts is different.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010731\/CNX_Precalc_Figure_03_04_0072.jpg\" alt=\"Graph of h(x)=x^3+4x^2+x-6.\" width=\"487\" height=\"329\" \/> <b>Figure 5.<\/b> Identifying the behavior of the graph at an x-intercept by examining the multiplicity of the zero.[\/caption]\r\n<p id=\"fs-id1165135407009\">The <em>x<\/em>-intercept [latex]x=-3[\/latex]\u00a0is the solution of equation [latex]x+3=0[\/latex]. The graph passes directly through the <em>x<\/em>-intercept at [latex]x=-3[\/latex]. The factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line\u2014it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.<\/p>\r\n<p id=\"fs-id1165137897788\">The <em>x<\/em>-intercept [latex]x=2[\/latex] is the repeated solution of the equation [latex]{\\left(x - 2\\right)}^{2}=0[\/latex]. The graph touches the axis at the intercept and changes direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic\u2014it bounces off of the horizontal axis at the intercept.<\/p>\r\n\r\n<div id=\"eip-608\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{\\left(x - 2\\right)}^{2}=\\left(x - 2\\right)\\left(x - 2\\right)[\/latex]<\/div>\r\n<p id=\"fs-id1165137888924\">The factor is repeated, that is, the factor [latex]\\left(x - 2\\right)[\/latex] appears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the <strong>multiplicity<\/strong>. The zero associated with this factor, [latex]x=2[\/latex], has multiplicity 2 because the factor [latex]\\left(x - 2\\right)[\/latex] occurs twice.<\/p>\r\n<p id=\"fs-id1165133402140\">The <em>x-<\/em>intercept [latex]x=-1[\/latex] is the repeated solution of factor [latex]{\\left(x+1\\right)}^{3}=0[\/latex]. The graph passes through the axis at the intercept, but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic\u2014with the same S-shape near the intercept as the toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex]. We call this a triple zero, or a zero with multiplicity 3.<\/p>\r\nFor <strong>zeros<\/strong> with even multiplicities, the graphs <em>touch<\/em> or are tangent to the <em>x<\/em>-axis. For zeros with odd multiplicities, the graphs <em>cross<\/em> or intersect the <em>x<\/em>-axis. See Figure 6\u00a0for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"874\"]<img src=\"https:\/\/cnx.org\/resources\/404d5117e8c2b2cc187c001d0fcf267e8d3c7bbf\/CNX_Precalc_Figure_03_04_008_fixed.jpg\" alt=\"Three graphs, left to right, with zeros of multiplicity 1, 2, and 3.\" width=\"874\" height=\"324\" \/> <b>Figure 6<\/b>[\/caption]\r\n<p id=\"fs-id1165133078115\">For higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the horizontal axis but, for each increasing even power, the graph will appear flatter as it approaches and leaves the <em>x<\/em>-axis.<\/p>\r\n<p id=\"fs-id1165133447988\">For higher odd powers, such as 5, 7, and 9, the graph will still cross through the horizontal axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the <em>x<\/em>-axis.<\/p>\r\n\r\n<div id=\"fs-id1165135620829\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Graphical Behavior of Polynomials at <em>x<\/em>-Intercepts<\/h3>\r\n<p id=\"fs-id1165134036762\">If a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], the behavior near the <em>x<\/em>-intercept <em>h\u00a0<\/em>is determined by the power <em>p<\/em>. We say that [latex]x=h[\/latex] is a zero of <strong>multiplicity<\/strong> <em>p<\/em>.<\/p>\r\n<p id=\"fs-id1165137647546\">The graph of a polynomial function will touch the <em>x<\/em>-axis at zeros with even multiplicities. The graph will cross the <em>x<\/em>-axis at zeros with odd multiplicities.<\/p>\r\n<p id=\"fs-id1165135195405\">The sum of the multiplicities is the degree of the polynomial function.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135195409\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135195416\">How To: Given a graph of a polynomial function of degree <i>n<\/i>, identify the zeros and their multiplicities.<\/h3>\r\n<ol id=\"fs-id1165135547216\">\r\n \t<li>If the graph crosses the <em>x<\/em>-axis and appears almost linear at the intercept, it is a single zero.<\/li>\r\n \t<li>If the graph touches the <em>x<\/em>-axis and bounces off of the axis, it is a zero with even multiplicity.<\/li>\r\n \t<li>If the graph crosses the <em>x<\/em>-axis at a zero, it is a zero with odd multiplicity.<\/li>\r\n \t<li>The sum of the multiplicities is <em>n<\/em>. This includes non-real zeros.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_04_06\" class=\"example\">\r\n<div id=\"fs-id1165137922408\" class=\"exercise\">\r\n<div id=\"fs-id1165135409401\" class=\"problem textbox shaded\">\r\n<h3>Example 7: Identifying Zeros and Their Multiplicities<\/h3>\r\nUse the graph of the function of degree 6 to identify the zeros of the function and their possible multiplicities.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010732\/CNX_Precalc_Figure_03_04_0092.jpg\" alt=\"Three graphs showing three different polynomial functions with multiplicity 1, 2, and 3.\" width=\"487\" height=\"628\" \/> <b>Figure 7<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"700901\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"700901\"]\r\n<p id=\"fs-id1165135533055\">The polynomial function is of degree <em>n<\/em>. The sum of the multiplicities must be <em>n<\/em>.<\/p>\r\n<p id=\"fs-id1165135641694\">Starting from the left, the first zero occurs at [latex]x=-3[\/latex]. The graph touches the <em>x<\/em>-axis, so the multiplicity of the zero must be even. The zero of \u20133 has multiplicity 2.<\/p>\r\n<p id=\"fs-id1165135369539\">The next zero occurs at [latex]x=-1[\/latex]. The graph looks almost linear at this point. This is a single zero of multiplicity 1.<\/p>\r\n<p id=\"fs-id1165135329820\">The last zero occurs at [latex]x=4[\/latex]. The graph crosses the<em> x<\/em>-axis, so the multiplicity of the zero must be odd. We know that the multiplicity is likely 3 and that the sum of the multiplicities is likely 6.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nUse the graph of the function of degree 5 to identify the zeros of the function and their multiplicities.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img class=\"small\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010732\/CNX_Precalc_Figure_03_04_0102.jpg\" alt=\"Graph of an even-degree polynomial with degree 6.\" width=\"487\" height=\"253\" \/> <b>Figure 8<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"166598\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"166598\"]\r\n\r\nThe graph has a zero of \u20135 with multiplicity 1, a zero of \u20131 with multiplicity 2, and a zero of 3 with even multiplicity.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"Example_03_03_08\" class=\"example\">\r\n<div id=\"fs-id1165137435581\" class=\"exercise\">\r\n<div id=\"fs-id1165137803210\" class=\"problem textbox shaded\">\r\n<h3>Example 8: Determining zeros and multiplicities of a Polynomial Function<\/h3>\r\n<p id=\"fs-id1165137441767\">Given the polynomial function [latex]f\\left(x\\right)=\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 4\\right)[\/latex], written in factored form for your convenience, determine the\u00a0zeros and multiplicities<\/p>\r\n[reveal-answer q=\"994834\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"994834\"]\r\n<p id=\"fs-id1165137863224\">The\u00a0zeros occur when the output is zero.<\/p>\r\n<p style=\"text-align: center;\">[latex](x - 2)^{3}(x+1)(x - 4)^{2}=0[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align} &amp;(x - 2)^{3}=0 &amp;&amp; \\text{or} &amp;&amp; x+1=0 &amp;&amp; \\text{or} &amp;&amp; (x - 4)^{2}=0 \\\\ &amp;x=2 &amp;&amp; \\text{or} &amp;&amp; x=-1 &amp;&amp; \\text{or} &amp;&amp; x=4 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165135316178\">The\u00a0zeros\u00a0are [latex]2,-1,4[\/latex].<\/p>\r\nThe multiplicities are the exponents on each of the factors:\u00a0 The zero 2 has a multiplicity of 3, the zero -1 has a multiplicity of 1, and the zero of 4 has a multiplicity of 2.\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h2>\u00a0Use factoring to \ufb01nd zeros of polynomial functions<\/h2>\r\n<h3>Find zeros of polynomial functions<\/h3>\r\n<p id=\"fs-id1165134042185\">Recall that if <em>f<\/em>\u00a0is a polynomial function, the values of <em>x<\/em>\u00a0for which [latex]f\\left(x\\right)=0[\/latex] are called <strong>zeros<\/strong> of <em>f<\/em>. If the equation of the polynomial function can be factored, we can set each factor equal to zero and solve for the zeros<strong>.<\/strong><\/p>\r\n<p id=\"fs-id1165134043725\">We can use this method to find <em>x<\/em>-intercepts because at the <em>x<\/em>-intercepts we find the input values when the output value is zero. For general polynomials, this can be a challenging prospect. While quadratics can be solved using the relatively simple quadratic formula, the corresponding formulas for cubic and fourth-degree polynomials are not simple enough to remember, and formulas do not exist for general higher-degree polynomials. Consequently, we will limit ourselves to three cases in this section:<\/p>\r\n\r\n<ol id=\"fs-id1165137733636\">\r\n \t<li>The polynomial can be factored using known methods: greatest common factor and trinomial factoring.<\/li>\r\n \t<li>The polynomial is given in factored form.<\/li>\r\n \t<li>Technology is used to determine the intercepts.<\/li>\r\n<\/ol>\r\n<div id=\"fs-id1165137640937\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137563367\">How To: Given a polynomial function <em>f<\/em>, find the <em>x<\/em>-intercepts by factoring.<\/h3>\r\n<ol id=\"fs-id1165134104993\">\r\n \t<li>Set [latex]f\\left(x\\right)=0[\/latex].<\/li>\r\n \t<li>If the polynomial function is not given in factored form:\r\n<ol id=\"fs-id1165137646354\">\r\n \t<li>Factor out any common monomial factors.<\/li>\r\n \t<li>Factor any factorable binomials or trinomials.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Set each factor equal to zero and solve to find the [latex]x\\text{-}[\/latex] intercepts.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_04_02\" class=\"example\">\r\n<div id=\"fs-id1165135191903\" class=\"exercise\">\r\n<div id=\"fs-id1165135179909\" class=\"problem textbox shaded\">\r\n<h3>Example 9: Finding the <em>x<\/em>-Intercepts of a Polynomial Function by Factoring<\/h3>\r\n<p id=\"fs-id1165137817691\">Find the <em>x<\/em>-intercepts of [latex]f\\left(x\\right)={x}^{6}-3{x}^{4}+2{x}^{2}[\/latex].<\/p>\r\n[reveal-answer q=\"546243\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"546243\"]\r\n<p id=\"fs-id1165137535791\">We can attempt to factor this polynomial to find solutions for [latex]f\\left(x\\right)=0[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align} &amp;{x}^{6}-3{x}^{4}+2{x}^{2}=0 &amp;&amp; \\\\ &amp;{x}^{2}\\left({x}^{4}-3{x}^{2}+2\\right)=0 &amp;&amp; \\text{Factor out the greatest common factor}. \\\\ &amp;{x}^{2}\\left({x}^{2}-1\\right)\\left({x}^{2}-2\\right)=0 &amp;&amp; \\text{Factor the trinomial}. \\\\ &amp;{x}^{2}\\left(x+1\\right)\\left(x-1\\right)\\left({x}^{2}-2\\right)=0 &amp;&amp; \\text{Factor the difference of squares}. \\end{align}[\/latex]<\/p>\r\nNow set each factor equal to zero and solve.\r\n<p style=\"text-align: center;\">[latex]\\begin{align} &amp; {x}^{2}=0 &amp;&amp; x+1=0 &amp;&amp; x-1=0 &amp;&amp; {x}^{2}-2=0 \\\\ &amp;x=0 &amp;&amp; x=-1 &amp;&amp; x=1 &amp;&amp; x=\\pm \\sqrt{2} \\end{align}[\/latex]<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010728\/CNX_Precalc_Figure_03_04_0032.jpg\" alt=\"Four graphs where the first graph is of an even-degree polynomial, the second graph is of an absolute function, the third graph is an odd-degree polynomial, and the fourth graph is a disjoint function.\" width=\"487\" height=\"224\" \/> <b>Figure 9<\/b>[\/caption]\r\n<p id=\"fs-id1165137932627\">This gives us five <em>x<\/em>-intercepts: [latex]\\left(0,0\\right),\\left(1,0\\right),\\left(-1,0\\right),\\left(\\sqrt{2},0\\right)[\/latex], and [latex]\\left(-\\sqrt{2},0\\right)[\/latex]. We can see that this is an even function.<\/p>\r\n[\/hidden-answer]<span id=\"fs-id1165134380378\">\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_04_03\" class=\"example\">\r\n<div id=\"fs-id1165137768835\" class=\"exercise\">\r\n<div id=\"fs-id1165137768837\" class=\"problem textbox shaded\">\r\n<h3>Example 10: Finding the <em>x<\/em>-Intercepts of a Polynomial Function by Factoring<\/h3>\r\n<p id=\"fs-id1165135254633\">Find the <em>x<\/em>-intercepts of [latex]f\\left(x\\right)={x}^{3}-5{x}^{2}-x+5[\/latex].<\/p>\r\n[reveal-answer q=\"996911\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"996911\"]\r\n<p id=\"fs-id1165137725387\">Find solutions for [latex]f\\left(x\\right)=0[\/latex]\u00a0by factoring.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align} &amp;{x}^{3}-5{x}^{2}-x+5=0 \\\\ &amp;{x}^{2}\\left(x - 5\\right)-1\\left(x - 5\\right)=0 &amp;&amp; \\text{Factor by grouping}. \\\\ &amp;\\left({x}^{2}-1\\right)\\left(x - 5\\right)=0 &amp;&amp; \\text{Factor out the common factor}. \\\\ &amp;\\left(x+1\\right)\\left(x - 1\\right)\\left(x - 5\\right)=0 &amp;&amp; \\text{Factor the difference of squares}. \\end{align}[\/latex]<\/p>\r\nNow we set each factor equal to 0.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;x+1=0 &amp;&amp; x - 1=0 &amp;&amp; x - 5=0 \\\\ &amp;x=-1 &amp;&amp; x=1 &amp;&amp; x=5 \\end{align}[\/latex]<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010728\/CNX_Precalc_Figure_03_04_0042.jpg\" alt=\"Graph of f(x)=x^6-3x^4+2x^2 with its five intercepts, (-sqrt(2), 0), (-1, 0), (0, 0), (1, 0), and (sqrt(2), 0).\" width=\"487\" height=\"402\" \/> <b>Figure 10<\/b>[\/caption]\r\n<p id=\"fs-id1165134541162\">There are three <em>x<\/em>-intercepts: [latex]\\left(-1,0\\right),\\left(1,0\\right)[\/latex], and [latex]\\left(5,0\\right)[\/latex].<\/p>\r\n[\/hidden-answer]<span id=\"fs-id1165133344112\">\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_04_04\" class=\"example\">\r\n<div id=\"fs-id1165135154515\" class=\"exercise\">\r\n<div id=\"fs-id1165135154517\" class=\"problem textbox shaded\">\r\n<h3>Example 11: Finding the <em>y<\/em>- and <em>x<\/em>-Intercepts of a Polynomial in Factored Form<\/h3>\r\n<p id=\"fs-id1165135528940\">Find the <i>y<\/i>-\u00a0and <em>x<\/em>-intercepts of [latex]g\\left(x\\right)={\\left(x - 2\\right)}^{2}\\left(2x+3\\right)[\/latex].<\/p>\r\n[reveal-answer q=\"180029\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"180029\"]\r\n<p id=\"fs-id1165135421555\">The <em>y<\/em>-intercept can be found by evaluating [latex]g\\left(0\\right)[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]g\\left(0\\right)={\\left(0 - 2\\right)}^{2}\\left(2\\left(0\\right)+3\\right)=12[\/latex]<\/p>\r\n<p id=\"eip-id1165134130215\">So the <em>y<\/em>-intercept is [latex]\\left(0,12\\right)[\/latex].<\/p>\r\n<p id=\"fs-id1165137870836\">The <em>x<\/em>-intercepts can be found by solving [latex]g\\left(x\\right)=0[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]{\\left(x - 2\\right)}^{2}\\left(2x+3\\right)=0[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{\\left(x - 2\\right)}^{2}=0 &amp;&amp; 2x+3=0 \\\\ &amp;x=2 &amp;&amp;x=-\\frac{3}{2} \\end{align}[\/latex]<\/p>\r\n<p id=\"eip-id1165135518219\">So the <em>x<\/em>-intercepts are [latex]\\left(2,0\\right)[\/latex] and [latex]\\left(-\\frac{3}{2},0\\right)[\/latex].<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nWe can always check that our answers are reasonable by using a graphing calculator to graph the polynomial as shown in Figure 11.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img class=\"small\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010731\/CNX_Precalc_Figure_03_04_0052.jpg\" alt=\"Graph of f(x)=x^3-5x^2-x+5 with its three intercepts (-1, 0), (1, 0), and (5, 0).\" width=\"487\" height=\"670\" \/> <b>Figure 11<\/b>[\/caption]\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_04_05\" class=\"example\">\r\n<div id=\"fs-id1165137415980\" class=\"exercise\">\r\n<div id=\"fs-id1165134381752\" class=\"problem textbox shaded\">\r\n<h3>Example 12: Finding the <em>x<\/em>-Intercepts of a Polynomial Function Using a Graph<\/h3>\r\n<p id=\"fs-id1165137453950\">Find the <em>x<\/em>-intercepts of [latex]h\\left(x\\right)={x}^{3}+4{x}^{2}+x - 6[\/latex].<\/p>\r\n[reveal-answer q=\"512408\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"512408\"]\r\n<p id=\"fs-id1165137895270\">This polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques previously discussed. Fortunately, we can use technology to find the intercepts. Keep in mind that some values make graphing difficult by hand. In these cases, we can take advantage of graphing utilities.<\/p>\r\nLooking at the graph of this function, as shown in Figure 12, it appears that there are <em>x<\/em>-intercepts at [latex]x=-3,-2[\/latex], and 1.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010731\/CNX_Precalc_Figure_03_04_0062.jpg\" alt=\"Graph of g(x)=(x-2)^2(2x+3) with its two x-intercepts (2, 0) and (-3\/2, 0) and its y-intercept (0, 12).\" width=\"487\" height=\"440\" \/> <b>Figure 12<\/b>[\/caption]\r\n<p id=\"fs-id1165131891784\">We can check whether these are correct by substituting these values for <em>x<\/em>\u00a0and verifying that the function is equal to 0.<\/p>\r\n<p id=\"fs-id1165135600839\">Since [latex]h\\left(x\\right)={x}^{3}+4{x}^{2}+x - 6[\/latex], we have:<\/p>\r\n<p style=\"text-align: center;\">[latex]h\\left(-3\\right)={\\left(-3\\right)}^{3}+4{\\left(-3\\right)}^{2}+\\left(-3\\right)-6=-27+36 - 3-6=0[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]h\\left(-2\\right)={\\left(-2\\right)}^{3}+4{\\left(-2\\right)}^{2}+\\left(-2\\right)-6=-8+16 - 2-6=0[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]h\\left(1\\right)={\\left(1\\right)}^{3}+4{\\left(1\\right)}^{2}+\\left(1\\right)-6=1+4+1 - 6=0[\/latex]<\/p>\r\n<p id=\"fs-id1165134129941\">Each <em>x<\/em>-intercept corresponds to a zero of the polynomial function and each zero yields a factor, so we can now write the polynomial in factored form.<\/p>\r\n<p style=\"text-align: center;\">[latex]h\\left(x\\right)={x}^{3}+4{x}^{2}+x - 6=\\left(x+3\\right)\\left(x+2\\right)\\left(x - 1\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165133238478\">Find the <em>y<\/em>-\u00a0and <em>x<\/em>-intercepts of the function [latex]f\\left(x\\right)={x}^{4}-19{x}^{2}+30x[\/latex].<\/p>\r\n[reveal-answer q=\"123401\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"123401\"]\r\n\r\ny-intercept [latex]\\left(0,0\\right)[\/latex]; x-intercepts [latex]\\left(0,0\\right),\\left(-5,0\\right),\\left(2,0\\right)[\/latex], and [latex]\\left(3,0\\right)[\/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\n[ohm_question hide_question_numbers=1]66678[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>\u00a0Determine end behavior<\/h2>\r\n<p id=\"fs-id1165135514626\">As we have already learned, the behavior of a graph of a <strong>polynomial function<\/strong> of the form<\/p>\r\n\r\n<div id=\"eip-263\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+...+{a}_{1}x+{a}_{0}[\/latex]<\/div>\r\n<p id=\"eip-id1165134547362\">will either ultimately rise or fall as <em>x<\/em>\u00a0increases without bound and will either rise or fall as <em>x\u00a0<\/em>decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say \u2013100 or \u20131,000.<\/p>\r\n<p id=\"fs-id1165132959259\">Recall that we call this behavior the <em>end behavior<\/em> of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function, [latex]{a}_{n}{x}^{n}[\/latex], is an even power function, as <em>x<\/em>\u00a0increases or decreases without bound, [latex]f\\left(x\\right)[\/latex] increases without bound. When the leading term is an odd power function, as\u00a0<em>x<\/em>\u00a0decreases without bound, [latex]f\\left(x\\right)[\/latex] also decreases without bound; as <em>x<\/em>\u00a0increases without bound, [latex]f\\left(x\\right)[\/latex] also increases without bound. If the leading term is negative, it will change the direction of the end behavior. The table below\u00a0summarizes all four cases.<\/p>\r\n\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Even Degree<\/th>\r\n<th>Odd Degree<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/11.png\"><img class=\"alignnone size-full wp-image-12504\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/11.png\" alt=\"11\" width=\"423\" height=\"559\" \/><\/a><\/td>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/12.png\"><img class=\"alignnone size-full wp-image-12505\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/12.png\" alt=\"12\" width=\"397\" height=\"560\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/13.png\"><img class=\"alignnone size-full wp-image-12506\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/13.png\" alt=\"13\" width=\"387\" height=\"574\" \/><\/a><\/td>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/14.png\"><img class=\"alignnone size-full wp-image-12507\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/14.png\" alt=\"14\" width=\"404\" height=\"564\" \/><\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Writing Formulas for Polynomial Functions<\/h2>\r\n<p id=\"fs-id1165135369122\">Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a <strong>polynomial function<\/strong> written in factored form will have an <em>x<\/em>-intercept where each factor is equal to zero, we can form a function that will pass through a set of <em>x<\/em>-intercepts by introducing a corresponding set of factors.<\/p>\r\n\r\n<div id=\"fs-id1165133320785\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Factored Form of Polynomials<\/h3>\r\n<p id=\"fs-id1165133320793\">If a polynomial of lowest degree <em>p<\/em>\u00a0has horizontal intercepts at [latex]x={x}_{1},{x}_{2},\\dots ,{x}_{n}[\/latex],\u00a0then the polynomial can be written in the factored form: [latex]f\\left(x\\right)=a{\\left(x-{x}_{1}\\right)}^{{p}_{1}}{\\left(x-{x}_{2}\\right)}^{{p}_{2}}\\cdots {\\left(x-{x}_{n}\\right)}^{{p}_{n}}[\/latex]\u00a0where the powers [latex]{p}_{i}[\/latex]\u00a0on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor <em>a<\/em>\u00a0can be determined given a value of the function other than the <em>x<\/em>-intercept.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135580289\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135580296\">How To: Given a graph of a polynomial function, write a formula for the function.<\/h3>\r\n<ol id=\"fs-id1165133309878\">\r\n \t<li>Identify the <em>x<\/em>-intercepts of the graph to find the factors of the polynomial.<\/li>\r\n \t<li>Examine the behavior of the graph at the <em>x<\/em>-intercepts to determine the multiplicity of each factor.<\/li>\r\n \t<li>Find the polynomial of least degree containing all the factors found in the previous step.<\/li>\r\n \t<li>Use any other point on the graph (the <em>y<\/em>-intercept may be easiest) to determine the stretch factor.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_04_10\" class=\"example\">\r\n<div id=\"fs-id1165134043949\" class=\"exercise\">\r\n<div id=\"fs-id1165134043951\" class=\"problem textbox shaded\">\r\n<h3>Example 13: Writing a Formula for a Polynomial Function from the Graph<\/h3>\r\nWrite a formula for the polynomial function shown in Figure 13.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010735\/CNX_Precalc_Figure_03_04_0242.jpg\" alt=\"Graph of a positive even-degree polynomial with zeros at x=-3, 2, 5 and y=-2.\" width=\"487\" height=\"366\" \/> <b>Figure 13<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"574656\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"574656\"]\r\n<p id=\"fs-id1165135621955\">his graph has three <em>x<\/em>-intercepts: <em>x\u00a0<\/em>= \u20133, 2, and 5. The <em>y<\/em>-intercept is located at (0, 2). At <em>x\u00a0<\/em>= \u20133 and <em>x\u00a0<\/em>= 5,\u00a0the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear. At <em>x\u00a0<\/em>= 2, the graph bounces at the intercept, suggesting the corresponding factor of the polynomial will be second degree (quadratic). Together, this gives us<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a\\left(x+3\\right){\\left(x - 2\\right)}^{2}\\left(x - 5\\right)[\/latex]<\/p>\r\n<p id=\"fs-id1165135575901\">To determine the stretch factor, we utilize another point on the graph. We will use the <em>y<\/em>-intercept (0, \u20132), to solve for <em>a<\/em>.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}f\\left(0\\right)&amp;=a\\left(0+3\\right){\\left(0 - 2\\right)}^{2}\\left(0 - 5\\right) \\\\ -2&amp;=a\\left(0+3\\right){\\left(0 - 2\\right)}^{2}\\left(0 - 5\\right) \\\\ -2&amp;=-60a \\\\ a&amp;=\\frac{1}{30} \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165133437286\">The graphed polynomial appears to represent the function [latex]f\\left(x\\right)=\\frac{1}{30}\\left(x+3\\right){\\left(x - 2\\right)}^{2}\\left(x - 5\\right)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nGiven the graph in Figure 14, write a formula for the function shown.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010735\/CNX_Precalc_Figure_03_04_0252.jpg\" alt=\"Graph of a negative even-degree polynomial with zeros at x=-1, 2, 4 and y=-4.\" width=\"487\" height=\"291\" \/> <b>Figure 14<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"412515\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"412515\"]\r\n\r\n[latex]f\\left(x\\right)=-\\frac{1}{8}{\\left(x - 2\\right)}^{3}{\\left(x+1\\right)}^{2}\\left(x - 4\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135440065\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It 11<\/h3>\r\n[ohm_question hide_question_numbers=1]15942[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Using Local and Global Extrema<\/h2>\r\n<p id=\"fs-id1165135440070\">With quadratics, we were able to algebraically find the maximum or minimum value of the function by finding the vertex. For general polynomials, finding these turning points is not possible without more advanced techniques from calculus. Even then, finding where extrema occur can still be algebraically challenging. For now, we will estimate the locations of turning points using technology to generate a graph.<\/p>\r\n<p id=\"fs-id1165135440077\">Each turning point represents a local minimum or maximum. Sometimes, a turning point is the highest or lowest point on the entire graph. In these cases, we say that the turning point is a <strong>global maximum <\/strong>or a <strong>global minimum<\/strong>. These are also referred to as the absolute maximum and absolute minimum values of the function.<\/p>\r\n\r\n<div id=\"fs-id1165133248530\" class=\"note\">\r\n<h3 class=\"title\">Local and Global Extrema<\/h3>\r\n<p id=\"fs-id1165133248538\">A <strong>local maximum<\/strong> or <strong>local minimum<\/strong> at <em>x\u00a0<\/em>= <em>a<\/em>\u00a0(sometimes called the relative maximum or minimum, respectively) is the output at the highest or lowest point on the graph in an open interval around <em>x\u00a0<\/em>= <em>a<\/em>. If a function has a local maximum at <em>a<\/em>, then [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex] for all <em>x<\/em>\u00a0in an open interval around <em>x<\/em> =\u00a0<em>a<\/em>. If a function has a local minimum at <em>a<\/em>, then [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex] for all <em>x<\/em>\u00a0in an open interval around <em>x\u00a0<\/em>= <em>a<\/em>.<\/p>\r\n<p id=\"fs-id1165134372821\">A <strong>global maximum<\/strong> or <strong>global minimum<\/strong> is the output at the highest or lowest point of the function. If a function has a global maximum at <em>a<\/em>, then [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex] for all <em>x<\/em>. If a function has a global minimum at <em>a<\/em>, then [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex] for all <em>x<\/em>.<\/p>\r\nWe can see the difference between local and global extrema in Figure 15.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010735\/CNX_Precalc_Figure_03_04_026n2.jpg\" alt=\"Graph of an even-degree polynomial that denotes the local maximum and minimum and the global maximum.\" width=\"487\" height=\"475\" \/> <b>Figure 15<\/b>[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135347671\" class=\"note precalculus qa textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<p id=\"fs-id1165134422158\"><strong>Do all polynomial functions have a global minimum or maximum?<\/strong><\/p>\r\n<p id=\"fs-id1165134422162\"><em>No. Only polynomial functions of even degree have a global minimum or maximum. For example, [latex]f\\left(x\\right)=x[\/latex] has neither a global maximum nor a global minimum.<\/em><\/p>\r\n\r\n<\/div>\r\n<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Key Equations<\/span>\r\n\r\n<section id=\"fs-id1165137724050\" class=\"key-equations\">\r\n<table id=\"eip-id1165134063974\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td>general form of a polynomial function<\/td>\r\n<td>[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section>\r\n<h2>Key Concepts<\/h2>\r\n<ul id=\"fs-id1165137846272\">\r\n \t<li>A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power.<\/li>\r\n \t<li>The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.<\/li>\r\n \t<li>A polynomial of degree <em>n<\/em>\u00a0will have at most <em>n<\/em>\u00a0<em>x-<\/em>intercepts and at most <em>n<\/em> \u2013 1\u00a0turning points.<\/li>\r\n \t<li>The graph of a polynomial function changes direction at its turning points.<\/li>\r\n \t<li>Polynomial functions of degree 2 or more are smooth, continuous functions.<\/li>\r\n \t<li>To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.<\/li>\r\n \t<li>Another way to find the <em>x-<\/em>intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the <em>x<\/em>-axis.<\/li>\r\n \t<li>The multiplicity of a zero determines how the graph behaves at the <em>x<\/em>-intercepts, in addition to the behavior equations at each zero.<\/li>\r\n \t<li>The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.<\/li>\r\n \t<li>The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.<\/li>\r\n \t<li>The end behavior of a polynomial function depends on the leading term.<\/li>\r\n<\/ul>\r\n<div><section id=\"fs-id1165137731646\" class=\"key-concepts\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165137668266\" class=\"definition\">\r\n \t<dt><strong>coefficient<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135194915\">a nonzero real number multiplied by a variable raised to an exponent<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135194918\" class=\"definition\">\r\n \t<dt><strong>continuous function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135194921\">a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137832108\" class=\"definition\">\r\n \t<dt><strong>degree<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137832112\">the highest power of the variable that occurs in a polynomial<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137832115\" class=\"definition\">\r\n \t<dt><strong>end behavior<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165131990654\">the behavior of the graph of a function as the input decreases without bound and increases without bound<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135347545\" class=\"definition\">\r\n \t<dt><strong>global maximum<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134043812\">highest turning point on a graph; [latex]f\\left(a\\right)[\/latex]\u00a0where [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex]\u00a0for all <em>x<\/em>.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131852045\" class=\"definition\">\r\n \t<dt><strong>global minimum<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165131852049\">lowest turning point on a graph; [latex]f\\left(a\\right)[\/latex]\u00a0where [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex]\r\nfor all <em>x<\/em>.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131990658\" class=\"definition\">\r\n \t<dt><strong>leading coefficient<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165131990661\">the coefficient of the leading term<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943522\" class=\"definition\">\r\n \t<dt><strong>leading term<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">the term containing the highest power of the variable<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134112772\" class=\"definition\">\r\n \t<dt><strong>multiplicity<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134112776\">the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], [latex]x=h[\/latex]\u00a0is a zero of multiplicity <em>p<\/em>.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943528\" class=\"definition\">\r\n \t<dt><strong>polynomial function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134297639\">a function that 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.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137644987\" class=\"definition\">\r\n \t<dt><strong>term of a polynomial function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137644990\">any [latex]{a}_{i}{x}^{i}[\/latex]\u00a0of a polynomial function in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133085661\" class=\"definition\">\r\n \t<dt><strong>turning point<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165133085665\">the location at which the graph of a function changes direction<\/dd>\r\n<\/dl>\r\n<\/section><\/div>\r\n<dl id=\"fs-id1165134112772\" class=\"definition\">\r\n \t<dd id=\"fs-id1165134112776\"><\/dd>\r\n<\/dl>\r\n&nbsp;\r\n<h2 style=\"text-align: center;\">Section 4.1 Homework Exercises<\/h2>\r\n1. What is the difference between an <em style=\"font-size: 1rem; font-weight: normal; text-align: initial; color: #373d3f;\">x<\/em><span style=\"font-size: 1rem; font-weight: normal; text-align: initial; color: #373d3f;\">-intercept and a zero of a polynomial function\u00a0[latex]y[\/latex]?<\/span>\r\n\r\n2.\u00a0If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?\r\n\r\n3.\u00a0What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?\r\n\r\n4. Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.\r\n\r\n5.\u00a0Explain how the factored form of the polynomial helps us in graphing it.\r\n\r\n6. If the graph of a polynomial just touches the <em>x<\/em>-axis and then changes direction, what can we conclude about the factored form of the polynomial?\r\n\r\nFor the following exercises, determine whether the function is polynomial function.\u00a0 If it is a polynomial function, indicate the degree and [latex]a_{n}[\/latex] (leading coefficient).\r\n\r\n7. [latex]\\text{ }f\\left(x\\right)=6{x}^{5}[\/latex]\r\n\r\n8. [latex]\\text{ }f\\left(x\\right)={\\left({x}^{2}\\right)}^{3}[\/latex]\r\n\r\n9. [latex]\\text{ }f\\left(x\\right)=\\frac{{x}^{2}}{{x}^{2}-1}[\/latex]\r\n\r\n10.\u00a0[latex]\\text{ }f\\left(x\\right)=x-{x}^{4}[\/latex]\r\n\r\n11.\u00a0[latex]\\text{ }f\\left(x\\right)=2x\\left(x+2\\right){\\left(x - 1\\right)}^{2}[\/latex]\r\n\r\n12. [latex]\\text{ }f\\left(x\\right)={3}^{x+1}[\/latex]\r\n\r\n13. [latex]\\text{ }f\\left(x\\right)=5\\sqrt(x)+2x^{4}[\/latex]\r\n\r\n14. [latex]\\text{ }f\\left(x\\right)=4x^{2}-\\dfrac{3}{x^{3}}[\/latex]\r\n\r\nFor the following exercises, find the intercepts of the functions.\r\n\r\n15. [latex]f\\left(t\\right)=2\\left(t - 1\\right)\\left(t+2\\right)\\left(t - 3\\right)[\/latex]\r\n\r\n16.\u00a0[latex]g\\left(n\\right)=-2\\left(3n - 1\\right)\\left(2n+1\\right)[\/latex]\r\n\r\n17.\u00a0[latex]f\\left(x\\right)=-4(x+2)(x-2)[\/latex]\r\n\r\n18.\u00a0[latex]f\\left(x\\right)=\\left(x+3\\right)\\left(4{x}^{2}-1\\right)[\/latex]\r\n\r\n19. [latex]f\\left(x\\right)=x\\left({x}^{2}-2x - 8\\right)[\/latex]\r\n\r\nFor the following exercises, determine the least possible degree of the polynomial function shown.\r\n\r\n20.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005215\/CNX_Precalc_Figure_03_03_201.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/>\r\n\r\n21.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005216\/CNX_Precalc_Figure_03_03_202.jpg\" alt=\"Graph of an even-degree polynomial.\" \/>\r\n\r\n22.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005216\/CNX_Precalc_Figure_03_03_203.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/>\r\n\r\n23.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005216\/CNX_Precalc_Figure_03_03_204.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/>\r\n\r\n24.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005216\/CNX_Precalc_Figure_03_03_205.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/>\r\n\r\n25.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005216\/CNX_Precalc_Figure_03_03_206.jpg\" alt=\"Graph of an even-degree polynomial.\" \/>\r\n\r\n26.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005217\/CNX_Precalc_Figure_03_03_207.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/>\r\n\r\nFor the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.\r\n\r\n27.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005217\/CNX_Precalc_Figure_03_03_209.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/>\r\n\r\n28.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005219\/CNX_Precalc_Figure_03_03_210.jpg\" alt=\"Graph of an equation.\" \/>\r\n\r\n29.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005220\/CNX_Precalc_Figure_03_03_211.jpg\" alt=\"Graph of an even-degree polynomial.\" \/>\r\n\r\n30.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005220\/CNX_Precalc_Figure_03_03_212.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/>\r\n\r\n31.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005221\/CNX_Precalc_Figure_03_03_213.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/>\r\n\r\n32.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005221\/CNX_Precalc_Figure_03_03_214.jpg\" alt=\"Graph of an equation.\" \/>\r\n\r\n33.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005221\/CNX_Precalc_Figure_03_03_215.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/>\r\n\r\nFor the following exercises, find the zeros and multiplicities.\r\n\r\n34. [latex]f\\left(x\\right)={\\left(x+2\\right)}^{3}{\\left(x - 3\\right)}^{2}[\/latex]\r\n\r\n35. [latex]f\\left(x\\right)={x}^{2}{\\left(2x+3\\right)}^{3}{\\left(x - 4\\right)}^{2}[\/latex]\r\n\r\n36.\u00a0[latex]f\\left(x\\right)={x}^{3}{\\left(x - 1\\right)}^{3}\\left(x+2\\right)[\/latex]\r\n\r\n37. [latex]f\\left(x\\right)={x}^{2}\\left({x}^{2}+4x+4\\right)[\/latex]\r\n\r\n38.\u00a0[latex]f\\left(x\\right)={\\left(2x+1\\right)}^{3}\\left(9{x}^{2}-6x+1\\right)[\/latex]\r\n\r\n39. [latex]f\\left(x\\right)={\\left(3x+2\\right)}^{3}\\left({x}^{2}-10x+25\\right)[\/latex]\r\n\r\n40.\u00a0[latex]f\\left(x\\right)=x\\left(4{x}^{2}-12x+9\\right)\\left({x}^{2}+8x+16\\right)[\/latex]\r\n\r\n41. [latex]f\\left(x\\right)={x}^{6}-{x}^{5}-2{x}^{4}[\/latex]\r\n\r\n32.\u00a0[latex]f\\left(x\\right)=3{x}^{4}+6{x}^{3}+3{x}^{2}[\/latex]\r\n\r\n43. [latex]f\\left(x\\right)=4{x}^{5}-12{x}^{4}+9{x}^{3}[\/latex]\r\n\r\n44.\u00a0[latex]f\\left(x\\right)=2{x}^{4}\\left({x}^{3}-4{x}^{2}+4x\\right)[\/latex]\r\n\r\n45. [latex]f\\left(x\\right)=4{x}^{4}\\left(9{x}^{4}-12{x}^{3}+4{x}^{2}\\right)[\/latex]\r\n\r\nFor the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or \u20131. There may be more than one correct answer.\r\n\r\n46. The <em>y<\/em>-intercept is [latex]\\left(0,-6\\right)[\/latex]. The <em>x<\/em>-intercepts are [latex]\\left(-3,0\\right),\\left(2,0\\right)[\/latex]. Degree is 2.\r\n\r\nEnd behavior: [latex]\\text{as }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty [\/latex].\r\n\r\n47. The <em>y<\/em>-intercept is [latex]\\left(0,-4\\right)[\/latex]. The <em>x<\/em>-intercepts are [latex]\\left(-2,0\\right),\\left(2,0\\right)[\/latex]. Degree is 2.\r\n\r\nEnd behavior: [latex]\\text{as }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty [\/latex].\r\n\r\n48.\u00a0The <em>y<\/em>-intercept is [latex]\\left(0,9\\right)[\/latex]. The <em>x<\/em>-intercepts are [latex]\\left(-3,0\\right),\\left(3,0\\right)[\/latex]. Degree is 2.\r\n\r\nEnd behavior: [latex]\\text{as }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to -\\infty [\/latex].\r\n\r\n49. The <em>y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex]. The <em>x<\/em>-intercepts are [latex]\\left(0,0\\right),\\left(2,0\\right)[\/latex]. Degree is 3.\r\n\r\nEnd behavior: [latex]\\text{as }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty [\/latex].\r\n\r\n50.\u00a0The <em>y-<\/em>intercept is [latex]\\left(0,1\\right)[\/latex]. The <em>x<\/em>-intercept is [latex]\\left(1,0\\right)[\/latex]. Degree is 3.\r\n\r\nEnd behavior: [latex]\\text{as }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to -\\infty [\/latex].\r\n\r\n51. The <em>y<\/em>-intercept is [latex]\\left(0,1\\right)[\/latex]. There is no <em>x<\/em>-intercept. Degree is 4.\r\n\r\nEnd behavior: [latex]\\text{as }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex].\r\n\r\nFor the following exercises, use the graphs to write the formula for a polynomial function of least degree.\r\n\r\n52.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005231\/CNX_Precalc_Figure_03_04_207.jpg\" alt=\"Graph of a positive odd-degree polynomial with zeros at x=-2, 1, and 3.\" \/>\r\n\r\n53.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005231\/CNX_PreCalc_Figure_03_04_208.jpg\" alt=\"Graph of a negative odd-degree polynomial with zeros at x=-3, 1, and 3.\" \/>\r\n54.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005232\/CNX_PreCalc_Figure_03_04_209.jpg\" alt=\"Graph of a negative odd-degree polynomial with zeros at x=-1, and 2.\" \/>\r\n\r\n55.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005232\/CNX_PreCalc_Figure_03_04_210.jpg\" alt=\"Graph of a positive odd-degree polynomial with zeros at x=-2, and 3.\" \/>\r\n\r\n56.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005232\/CNX_PreCalc_Figure_03_04_211.jpg\" alt=\"Graph of a negative even-degree polynomial with zeros at x=-3, -2, 3, and 4.\" \/>\r\n\r\n57.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005232\/CNX_PreCalc_Figure_03_04_212.jpg\" alt=\"Graph of a negative even-degree polynomial with zeros at x=-4, -2, 1, and 3.\" \/>\r\n58.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005233\/CNX_PreCalc_Figure_03_04_213.jpg\" alt=\"Graph of a positive even-degree polynomial with zeros at x=-4, -2, and 3.\" \/>\r\n\r\n59.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005233\/CNX_PreCalc_Figure_03_04_215.jpg\" alt=\"Graph of a negative odd-degree polynomial with zeros at x=-3, -2, and 1.\" \/>\r\n\r\n<\/section><\/section>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify polynomial functions<\/li>\n<li>Identify the degree, leading coefficient, and multiplicity of polynomial functions<\/li>\n<li>Identify end behavior of polynomial functions.<\/li>\n<li>Writing a formula for a polynomial function from the graph<\/li>\n<\/ul>\n<\/div>\n<h2>\u00a0Identify polynomial functions<\/h2>\n<p id=\"fs-id1165135689465\">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<div id=\"eip-719\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]r\\left(w\\right)=24+8w[\/latex]<\/div>\n<p id=\"fs-id1165133432974\">We can combine this with the formula for the area <em>A<\/em>\u00a0of a circle.<\/p>\n<div id=\"eip-731\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/div>\n<p id=\"fs-id1165137704887\">Composing these functions gives a formula for the area in terms of weeks.<\/p>\n<div id=\"eip-645\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}A\\left(w\\right)&=A\\left(r\\left(w\\right)\\right)\\\\ &=A\\left(24+8w\\right)\\\\ &=\\pi {\\left(24+8w\\right)}^{2}\\end{align}[\/latex]<\/div>\n<p id=\"fs-id1165137835475\">Multiplying gives the formula.<\/p>\n<div id=\"eip-290\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A\\left(w\\right)=576\\pi +384\\pi w+64\\pi {w}^{2}[\/latex]<\/div>\n<p id=\"fs-id1165135205726\">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 id=\"fs-id1165137715427\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Polynomial Functions<\/h3>\n<p id=\"fs-id1165137823247\">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 id=\"eip-id1165137832690\">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 id=\"fs-id1165137715427\" class=\"note textbox\">\n<h3 class=\"title\">Characteristics of Polynomial Functions<\/h3>\n<ul>\n<li>A polynomial function consists of either zero or the sum of a finite number of non-zero\u00a0terms<\/li>\n<li>A polynomial must have whole number exponents [latex](0,1,2,3,...)[\/latex]<\/li>\n<li>A polynomial cannot have negative, fractional, or decimal exponents<\/li>\n<li>A polynomial must be a smooth continuous curve with no breaks or cusps<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_03_03_04\" class=\"example\">\n<div id=\"fs-id1165137817691\" class=\"exercise\">\n<div id=\"fs-id1165137817693\" class=\"problem textbox shaded\">\n<h3>Example 1: Identifying Polynomial Functions<\/h3>\n<p id=\"fs-id1165135262000\">Which of the following are polynomial functions?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}f\\left(x\\right)=2{x}^{3}\\cdot 3x+4 \\\\ g\\left(x\\right)=-x\\left({x}^{2}-4\\right) \\\\ h\\left(x\\right)=5\\sqrt{x}+2 \\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q824812\">Show Solution<\/span><\/p>\n<div id=\"q824812\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134094645\">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 id=\"fs-id1165137864157\">\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.\u00a0 In addition, the exponent is [latex]\\frac{1}{2}[\/latex], which means it is not a polynomial.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>\u00a0Identify the degree and leading coefficient of polynomial functions<\/h2>\n<p id=\"fs-id1165137831216\">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 id=\"fs-id1165135193124\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Terminology of Polynomial Functions<\/h3>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010717\/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 class=\"wp-caption-text\"><b>Figure 1<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137921667\">We often rearrange polynomials so that the powers are descending.<span id=\"fs-id1165137406148\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165137482568\">When a polynomial is written in this way, we say that it is in general form.<\/p>\n<\/div>\n<div id=\"fs-id1165134031372\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137803898\">How To: Given a polynomial function, identify the degree and leading coefficient.<\/h3>\n<ol id=\"fs-id1165135587816\">\n<li>Find the highest power of <em>x\u00a0<\/em>to determine the degree function.<\/li>\n<li>Identify the term containing the highest power of <em>x\u00a0<\/em>to find the leading term.<\/li>\n<li>Identify the coefficient of the leading term.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_03_05\" class=\"example\">\n<div id=\"fs-id1165137401820\" class=\"exercise\">\n<div id=\"fs-id1165137862379\" class=\"problem textbox shaded\">\n<h3>Example 2: Identifying the Degree and Leading Coefficient of a Polynomial Function<\/h3>\n<p id=\"fs-id1165137435372\">Identify the degree, leading term, and leading coefficient of the following polynomial functions.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered} 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{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q951580\">Show Solution<\/span><\/p>\n<div id=\"q951580\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137722510\">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, \u20134.<\/p>\n<p id=\"fs-id1165135457771\">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 id=\"fs-id1165135503949\">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, \u20131.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137424484\">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=\"q148647\">Show Solution<\/span><\/p>\n<div id=\"q148647\" class=\"hidden-answer\" style=\"display: none\">\n<p>The degree is 6. The leading term is [latex]-{x}^{6}[\/latex]. The leading coefficient is \u20131.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm34293\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=34293&theme=oea&iframe_resize_id=ohm34293\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Identifying Local Behavior of Polynomial Functions<\/h2>\n<p id=\"fs-id1165134054039\">A <strong>turning point <\/strong>is a point at which the function values change from increasing to decreasing or decreasing to increasing.<\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010719\/CNX_Precalc_Figure_03_03_0172.jpg\" alt=\"\" width=\"731\" height=\"629\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<h2>Understand the relationship between degree and turning points<\/h2>\n<p id=\"fs-id1165135416524\">\u00a0Look at the graph of the polynomial function [latex]f\\left(x\\right)={x}^{4}-{x}^{3}-4{x}^{2}+4x[\/latex] in Figure 3. The graph has three turning points.<span id=\"fs-id1165134155116\"><br \/>\n<\/span><\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010733\/CNX_Precalc_Figure_03_04_0152.jpg\" alt=\"Graph of an odd-degree polynomial with a negative leading coefficient. Note that as x goes to positive infinity, f(x) goes to negative infinity, and as x goes to negative infinity, f(x) goes to positive infinity.\" width=\"487\" height=\"327\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137784439\">This function <em>f<\/em>\u00a0is a 4<sup>th<\/sup> degree polynomial function and has 3 turning points. The maximum number of turning points of a polynomial function is always one less than the degree of the function.<\/p>\n<div id=\"fs-id1165135502799\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Interpreting Turning Points<\/h3>\n<p id=\"fs-id1165135469050\">A <strong>turning point<\/strong> is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising).<\/p>\n<p id=\"fs-id1165135469055\">A polynomial of degree <em>n<\/em>\u00a0will have at most <em>n<\/em> \u2013 1\u00a0turning points.<\/p>\n<\/div>\n<div id=\"Example_03_04_07\" class=\"example\">\n<div id=\"fs-id1165134374690\" class=\"exercise\">\n<div id=\"fs-id1165134060420\" class=\"problem textbox shaded\">\n<h3>Example 3: Finding the Maximum Number of Turning Points Using the Degree of a Polynomial Function<\/h3>\n<p id=\"fs-id1165134060425\">Find the maximum number of turning points of each polynomial function.<\/p>\n<ol id=\"fs-id1165134060428\">\n<li>[latex]f\\left(x\\right)=-{x}^{3}+4{x}^{5}-3{x}^{2}++1[\/latex]<\/li>\n<li>[latex]f\\left(x\\right)=-{\\left(x - 1\\right)}^{2}\\left(1+2{x}^{2}\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q157524\">Show Solution<\/span><\/p>\n<div id=\"q157524\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165137784430\">\n<li>[latex]f\\left(x\\right)=-x{}^{3}+4{x}^{5}-3{x}^{2}++1[\/latex]\n<p id=\"fs-id1165135335895\">First, rewrite the polynomial function in descending order: [latex]f\\left(x\\right)=4{x}^{5}-{x}^{3}-3{x}^{2}++1[\/latex]<\/p>\n<p id=\"fs-id1165135453844\">Identify the degree of the polynomial function. This polynomial function is of degree 5.<\/p>\n<p id=\"fs-id1165135341233\">The maximum number of turning points is 5 \u2013 1 = 4.<\/p>\n<\/li>\n<li>[latex]f\\left(x\\right)=-{\\left(x - 1\\right)}^{2}\\left(1+2{x}^{2}\\right)[\/latex]<\/li>\n<\/ol>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2019\/04\/01021335\/CNX_Precalc_Figure_03_04_0162.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-15117 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2019\/04\/01021335\/CNX_Precalc_Figure_03_04_0162.jpg\" alt=\"Graphic of f(x) showing to multiply the first term of (x-1)^2 and 2x^2 to determine the leading term.\" width=\"487\" height=\"67\" \/><\/a><\/p>\n<p style=\"text-align: center;\">[latex]a_{n}=-\\left(x^2\\right)\\left(2x^2\\right)=-2x^4[\/latex]<\/p>\n<p id=\"fs-id1165133104532\">First, identify the leading term of the polynomial function if the function were expanded.<span id=\"fs-id1165134130071\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165135551181\">Then, identify the degree of the polynomial function. This polynomial function is of degree 4.<\/p>\n<p id=\"fs-id1165135551185\">The maximum number of turning points is 4 \u2013 1 = 3.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135188274\">Without graphing the function, determine the maximum number of\u00a0turning points for [latex]f\\left(x\\right)=108 - 13{x}^{9}-8{x}^{4}+14{x}^{12}+2{x}^{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q515707\">Show Solution<\/span><\/p>\n<div id=\"q515707\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are at most\u00a011 turning points.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Comparing Smooth and Continuous Graphs<\/span><\/p>\n<section id=\"fs-id1165134080932\">\n<p id=\"fs-id1165137692509\">The degree of a polynomial function helps us to determine\u00a0the number of turning points. A polynomial function of\u00a0<em>n<\/em>th degree is the product of <em>n<\/em>\u00a0factors, so it will have at most <em>n<\/em>\u00a0roots or zeros, or <em>x<\/em>-intercepts. The graph of the polynomial function of degree <em>n<\/em>\u00a0must have at most <em>n<\/em> \u2013 1\u00a0turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.<\/p>\n<p id=\"fs-id1165137657937\">A <strong>continuous function<\/strong> has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A <strong>smooth curve<\/strong> is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.<\/p>\n<div id=\"Example_03_04_01\" class=\"example\">\n<div id=\"fs-id1165137643218\" class=\"exercise\">\n<div id=\"fs-id1165133360328\" class=\"problem textbox shaded\">\n<h3>Example 4: Recognizing Polynomial Functions<\/h3>\n<p>Which of the graphs in Figure 4\u00a0represents a degree 3 polynomial function?<\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010727\/CNX_Precalc_Figure_03_04_0022.jpg\" alt=\"Two graphs in which one has a polynomial function and the other has a function closely resembling a polynomial but is not.\" width=\"731\" height=\"766\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 4<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q898519\">Show Solution<\/span><\/p>\n<div id=\"q898519\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134188794\">The graphs of <em>g<\/em>\u00a0and <em>k\u00a0<\/em>are graphs of functions that are not polynomials. The graph of function <em>g<\/em>\u00a0has a sharp corner. The graph of function <em>k<\/em>\u00a0is not continuous.<\/p>\n<p>The graph of f is a polynomial, however it has 4 turning points, which is too many for a degree 3 polynomial.<\/p>\n<p>The graph if h could be a degree 3 polynomial.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<section id=\"fs-id1165137735781\">\n<div id=\"fs-id1165137766902\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137645233\">The Factor Theorem<\/h3>\n<p>For the polynomial [latex]P(x)[\/latex],<\/p>\n<ol id=\"fs-id1165137571388\">\n<li>If [latex]r[\/latex] is a zero of [latex]P(x)[\/latex] then [latex]x-r[\/latex] will be a factor of [latex]P(x)[\/latex].<\/li>\n<li>If [latex]x-r[\/latex] is a factor of [latex]P(x)[\/latex] then [latex]r[\/latex] will be a zero of [latex]P(x)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_03_08\" class=\"example\">\n<div id=\"fs-id1165137435581\" class=\"exercise\">\n<div id=\"fs-id1165137803210\" class=\"problem textbox shaded\">\n<h3>Example 5: Form a polynomial from given zeros<\/h3>\n<p>Given [latex]f(x)[\/latex] has zeros of [latex]-5,0,5[\/latex], form a degree three polynomial with integer coefficients.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q994840\">Show Solution<\/span><\/p>\n<div id=\"q994840\" class=\"hidden-answer\" style=\"display: none\">\nTo form the polynomial, we will use part 2 of the factor theorem. We will use the formula [latex]f(x)=\\left(x-r_{1}\\right)\\left(x-r_{2}\\right)\\left(x-r_{3}\\right)... \\\\ f(x)=\\left(x-(-5)\\right)\\left(x-0\\right)\\left(x-5\\right) \\\\ f(x)=x(x+5)(x-5) \\\\ f(x) = x^{3}-25x[\/latex]\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_08\" class=\"example\">\n<div id=\"fs-id1165137435581\" class=\"exercise\">\n<div id=\"fs-id1165137803210\" class=\"problem textbox shaded\">\n<h3>Example 6: Form a polynomial from given zeros<\/h3>\n<p>Given [latex]f(x)[\/latex] has zeros of [latex]-3,2[\/latex], form a degree three polynomial with integer coefficients.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q994850\">Show Solution<\/span><\/p>\n<div id=\"q994850\" class=\"hidden-answer\" style=\"display: none\">\nTo form the polynomial, we will use part 2 of the factor theorem. We will use the formula: [latex]f(x)=\\left(x-r_{1}\\right)\\left(x-r_{2}\\right)\\left(x-r_{3}\\right)... \\\\ f(x)=\\left(x-(-3)\\right)\\left(x-2\\right) \\\\ f(x)=(x+3)(x-2)[\/latex]<\/p>\n<p>However, [latex]f(x)=(x+3)(x-2)[\/latex] is not a degree 3 polynomial. To make it a degree three, [latex]f(x)[\/latex] can be written two different ways:<br \/>\n[latex]f(x)=(x+3)^{2}(x-2)\\text{ or} \\\\ f(x)=(x+3)(x-2)^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>In the previous example, writing a degree polynomial with zeros of [latex]-3,2[\/latex] resulted in two answers: [latex]f(x)=(x+3)^{2}(x-2)[\/latex] or [latex]f(x)=(x+3)(x-2)^{2}[\/latex]. The exponents on each of the factors are called <strong>multiplicities<\/strong>.<\/p>\n<p>Multiplicities are useful because they indicate whether a graph touches or crosses the x-axis.<\/p>\n<h2>Identify zeros and their multiplicities<\/h2>\n<p id=\"fs-id1165135581073\">Graphs behave differently at various <em>x<\/em>-intercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and bounce off.<\/p>\n<p id=\"fs-id1165133092720\">Suppose, for example, we graph the function<\/p>\n<div id=\"eip-840\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=\\left(x+3\\right){\\left(x - 2\\right)}^{2}{\\left(x+1\\right)}^{3}[\/latex].<\/div>\n<p>Notice in Figure 5\u00a0that the behavior of the function at each of the <em>x<\/em>-intercepts is different.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010731\/CNX_Precalc_Figure_03_04_0072.jpg\" alt=\"Graph of h(x)=x^3+4x^2+x-6.\" width=\"487\" height=\"329\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 5.<\/b> Identifying the behavior of the graph at an x-intercept by examining the multiplicity of the zero.<\/p>\n<\/div>\n<p id=\"fs-id1165135407009\">The <em>x<\/em>-intercept [latex]x=-3[\/latex]\u00a0is the solution of equation [latex]x+3=0[\/latex]. The graph passes directly through the <em>x<\/em>-intercept at [latex]x=-3[\/latex]. The factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line\u2014it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.<\/p>\n<p id=\"fs-id1165137897788\">The <em>x<\/em>-intercept [latex]x=2[\/latex] is the repeated solution of the equation [latex]{\\left(x - 2\\right)}^{2}=0[\/latex]. The graph touches the axis at the intercept and changes direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic\u2014it bounces off of the horizontal axis at the intercept.<\/p>\n<div id=\"eip-608\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{\\left(x - 2\\right)}^{2}=\\left(x - 2\\right)\\left(x - 2\\right)[\/latex]<\/div>\n<p id=\"fs-id1165137888924\">The factor is repeated, that is, the factor [latex]\\left(x - 2\\right)[\/latex] appears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the <strong>multiplicity<\/strong>. The zero associated with this factor, [latex]x=2[\/latex], has multiplicity 2 because the factor [latex]\\left(x - 2\\right)[\/latex] occurs twice.<\/p>\n<p id=\"fs-id1165133402140\">The <em>x-<\/em>intercept [latex]x=-1[\/latex] is the repeated solution of factor [latex]{\\left(x+1\\right)}^{3}=0[\/latex]. The graph passes through the axis at the intercept, but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic\u2014with the same S-shape near the intercept as the toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex]. We call this a triple zero, or a zero with multiplicity 3.<\/p>\n<p>For <strong>zeros<\/strong> with even multiplicities, the graphs <em>touch<\/em> or are tangent to the <em>x<\/em>-axis. For zeros with odd multiplicities, the graphs <em>cross<\/em> or intersect the <em>x<\/em>-axis. See Figure 6\u00a0for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.<\/p>\n<div style=\"width: 884px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/404d5117e8c2b2cc187c001d0fcf267e8d3c7bbf\/CNX_Precalc_Figure_03_04_008_fixed.jpg\" alt=\"Three graphs, left to right, with zeros of multiplicity 1, 2, and 3.\" width=\"874\" height=\"324\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165133078115\">For higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the horizontal axis but, for each increasing even power, the graph will appear flatter as it approaches and leaves the <em>x<\/em>-axis.<\/p>\n<p id=\"fs-id1165133447988\">For higher odd powers, such as 5, 7, and 9, the graph will still cross through the horizontal axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the <em>x<\/em>-axis.<\/p>\n<div id=\"fs-id1165135620829\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Graphical Behavior of Polynomials at <em>x<\/em>-Intercepts<\/h3>\n<p id=\"fs-id1165134036762\">If a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], the behavior near the <em>x<\/em>-intercept <em>h\u00a0<\/em>is determined by the power <em>p<\/em>. We say that [latex]x=h[\/latex] is a zero of <strong>multiplicity<\/strong> <em>p<\/em>.<\/p>\n<p id=\"fs-id1165137647546\">The graph of a polynomial function will touch the <em>x<\/em>-axis at zeros with even multiplicities. The graph will cross the <em>x<\/em>-axis at zeros with odd multiplicities.<\/p>\n<p id=\"fs-id1165135195405\">The sum of the multiplicities is the degree of the polynomial function.<\/p>\n<\/div>\n<div id=\"fs-id1165135195409\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135195416\">How To: Given a graph of a polynomial function of degree <i>n<\/i>, identify the zeros and their multiplicities.<\/h3>\n<ol id=\"fs-id1165135547216\">\n<li>If the graph crosses the <em>x<\/em>-axis and appears almost linear at the intercept, it is a single zero.<\/li>\n<li>If the graph touches the <em>x<\/em>-axis and bounces off of the axis, it is a zero with even multiplicity.<\/li>\n<li>If the graph crosses the <em>x<\/em>-axis at a zero, it is a zero with odd multiplicity.<\/li>\n<li>The sum of the multiplicities is <em>n<\/em>. This includes non-real zeros.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_04_06\" class=\"example\">\n<div id=\"fs-id1165137922408\" class=\"exercise\">\n<div id=\"fs-id1165135409401\" class=\"problem textbox shaded\">\n<h3>Example 7: Identifying Zeros and Their Multiplicities<\/h3>\n<p>Use the graph of the function of degree 6 to identify the zeros of the function and their possible multiplicities.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010732\/CNX_Precalc_Figure_03_04_0092.jpg\" alt=\"Three graphs showing three different polynomial functions with multiplicity 1, 2, and 3.\" width=\"487\" height=\"628\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q700901\">Show Solution<\/span><\/p>\n<div id=\"q700901\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135533055\">The polynomial function is of degree <em>n<\/em>. The sum of the multiplicities must be <em>n<\/em>.<\/p>\n<p id=\"fs-id1165135641694\">Starting from the left, the first zero occurs at [latex]x=-3[\/latex]. The graph touches the <em>x<\/em>-axis, so the multiplicity of the zero must be even. The zero of \u20133 has multiplicity 2.<\/p>\n<p id=\"fs-id1165135369539\">The next zero occurs at [latex]x=-1[\/latex]. The graph looks almost linear at this point. This is a single zero of multiplicity 1.<\/p>\n<p id=\"fs-id1165135329820\">The last zero occurs at [latex]x=4[\/latex]. The graph crosses the<em> x<\/em>-axis, so the multiplicity of the zero must be odd. We know that the multiplicity is likely 3 and that the sum of the multiplicities is likely 6.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Use the graph of the function of degree 5 to identify the zeros of the function and their multiplicities.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"small\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010732\/CNX_Precalc_Figure_03_04_0102.jpg\" alt=\"Graph of an even-degree polynomial with degree 6.\" width=\"487\" height=\"253\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 8<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q166598\">Show Solution<\/span><\/p>\n<div id=\"q166598\" class=\"hidden-answer\" style=\"display: none\">\n<p>The graph has a zero of \u20135 with multiplicity 1, a zero of \u20131 with multiplicity 2, and a zero of 3 with even multiplicity.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_08\" class=\"example\">\n<div id=\"fs-id1165137435581\" class=\"exercise\">\n<div id=\"fs-id1165137803210\" class=\"problem textbox shaded\">\n<h3>Example 8: Determining zeros and multiplicities of a Polynomial Function<\/h3>\n<p id=\"fs-id1165137441767\">Given the polynomial function [latex]f\\left(x\\right)=\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 4\\right)[\/latex], written in factored form for your convenience, determine the\u00a0zeros and multiplicities<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q994834\">Show Solution<\/span><\/p>\n<div id=\"q994834\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137863224\">The\u00a0zeros occur when the output is zero.<\/p>\n<p style=\"text-align: center;\">[latex](x - 2)^{3}(x+1)(x - 4)^{2}=0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &(x - 2)^{3}=0 && \\text{or} && x+1=0 && \\text{or} && (x - 4)^{2}=0 \\\\ &x=2 && \\text{or} && x=-1 && \\text{or} && x=4 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165135316178\">The\u00a0zeros\u00a0are [latex]2,-1,4[\/latex].<\/p>\n<p>The multiplicities are the exponents on each of the factors:\u00a0 The zero 2 has a multiplicity of 3, the zero -1 has a multiplicity of 1, and the zero of 4 has a multiplicity of 2.<\/p>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>\u00a0Use factoring to \ufb01nd zeros of polynomial functions<\/h2>\n<h3>Find zeros of polynomial functions<\/h3>\n<p id=\"fs-id1165134042185\">Recall that if <em>f<\/em>\u00a0is a polynomial function, the values of <em>x<\/em>\u00a0for which [latex]f\\left(x\\right)=0[\/latex] are called <strong>zeros<\/strong> of <em>f<\/em>. If the equation of the polynomial function can be factored, we can set each factor equal to zero and solve for the zeros<strong>.<\/strong><\/p>\n<p id=\"fs-id1165134043725\">We can use this method to find <em>x<\/em>-intercepts because at the <em>x<\/em>-intercepts we find the input values when the output value is zero. For general polynomials, this can be a challenging prospect. While quadratics can be solved using the relatively simple quadratic formula, the corresponding formulas for cubic and fourth-degree polynomials are not simple enough to remember, and formulas do not exist for general higher-degree polynomials. Consequently, we will limit ourselves to three cases in this section:<\/p>\n<ol id=\"fs-id1165137733636\">\n<li>The polynomial can be factored using known methods: greatest common factor and trinomial factoring.<\/li>\n<li>The polynomial is given in factored form.<\/li>\n<li>Technology is used to determine the intercepts.<\/li>\n<\/ol>\n<div id=\"fs-id1165137640937\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137563367\">How To: Given a polynomial function <em>f<\/em>, find the <em>x<\/em>-intercepts by factoring.<\/h3>\n<ol id=\"fs-id1165134104993\">\n<li>Set [latex]f\\left(x\\right)=0[\/latex].<\/li>\n<li>If the polynomial function is not given in factored form:\n<ol id=\"fs-id1165137646354\">\n<li>Factor out any common monomial factors.<\/li>\n<li>Factor any factorable binomials or trinomials.<\/li>\n<\/ol>\n<\/li>\n<li>Set each factor equal to zero and solve to find the [latex]x\\text{-}[\/latex] intercepts.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_04_02\" class=\"example\">\n<div id=\"fs-id1165135191903\" class=\"exercise\">\n<div id=\"fs-id1165135179909\" class=\"problem textbox shaded\">\n<h3>Example 9: Finding the <em>x<\/em>-Intercepts of a Polynomial Function by Factoring<\/h3>\n<p id=\"fs-id1165137817691\">Find the <em>x<\/em>-intercepts of [latex]f\\left(x\\right)={x}^{6}-3{x}^{4}+2{x}^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q546243\">Show Solution<\/span><\/p>\n<div id=\"q546243\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137535791\">We can attempt to factor this polynomial to find solutions for [latex]f\\left(x\\right)=0[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &{x}^{6}-3{x}^{4}+2{x}^{2}=0 && \\\\ &{x}^{2}\\left({x}^{4}-3{x}^{2}+2\\right)=0 && \\text{Factor out the greatest common factor}. \\\\ &{x}^{2}\\left({x}^{2}-1\\right)\\left({x}^{2}-2\\right)=0 && \\text{Factor the trinomial}. \\\\ &{x}^{2}\\left(x+1\\right)\\left(x-1\\right)\\left({x}^{2}-2\\right)=0 && \\text{Factor the difference of squares}. \\end{align}[\/latex]<\/p>\n<p>Now set each factor equal to zero and solve.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} & {x}^{2}=0 && x+1=0 && x-1=0 && {x}^{2}-2=0 \\\\ &x=0 && x=-1 && x=1 && x=\\pm \\sqrt{2} \\end{align}[\/latex]<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010728\/CNX_Precalc_Figure_03_04_0032.jpg\" alt=\"Four graphs where the first graph is of an even-degree polynomial, the second graph is of an absolute function, the third graph is an odd-degree polynomial, and the fourth graph is a disjoint function.\" width=\"487\" height=\"224\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 9<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137932627\">This gives us five <em>x<\/em>-intercepts: [latex]\\left(0,0\\right),\\left(1,0\\right),\\left(-1,0\\right),\\left(\\sqrt{2},0\\right)[\/latex], and [latex]\\left(-\\sqrt{2},0\\right)[\/latex]. We can see that this is an even function.<\/p>\n<\/div>\n<\/div>\n<p><span id=\"fs-id1165134380378\"><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_04_03\" class=\"example\">\n<div id=\"fs-id1165137768835\" class=\"exercise\">\n<div id=\"fs-id1165137768837\" class=\"problem textbox shaded\">\n<h3>Example 10: Finding the <em>x<\/em>-Intercepts of a Polynomial Function by Factoring<\/h3>\n<p id=\"fs-id1165135254633\">Find the <em>x<\/em>-intercepts of [latex]f\\left(x\\right)={x}^{3}-5{x}^{2}-x+5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q996911\">Show Solution<\/span><\/p>\n<div id=\"q996911\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137725387\">Find solutions for [latex]f\\left(x\\right)=0[\/latex]\u00a0by factoring.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &{x}^{3}-5{x}^{2}-x+5=0 \\\\ &{x}^{2}\\left(x - 5\\right)-1\\left(x - 5\\right)=0 && \\text{Factor by grouping}. \\\\ &\\left({x}^{2}-1\\right)\\left(x - 5\\right)=0 && \\text{Factor out the common factor}. \\\\ &\\left(x+1\\right)\\left(x - 1\\right)\\left(x - 5\\right)=0 && \\text{Factor the difference of squares}. \\end{align}[\/latex]<\/p>\n<p>Now we set each factor equal to 0.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&x+1=0 && x - 1=0 && x - 5=0 \\\\ &x=-1 && x=1 && x=5 \\end{align}[\/latex]<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010728\/CNX_Precalc_Figure_03_04_0042.jpg\" alt=\"Graph of f(x)=x^6-3x^4+2x^2 with its five intercepts, (-sqrt(2), 0), (-1, 0), (0, 0), (1, 0), and (sqrt(2), 0).\" width=\"487\" height=\"402\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 10<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165134541162\">There are three <em>x<\/em>-intercepts: [latex]\\left(-1,0\\right),\\left(1,0\\right)[\/latex], and [latex]\\left(5,0\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><span id=\"fs-id1165133344112\"><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_04_04\" class=\"example\">\n<div id=\"fs-id1165135154515\" class=\"exercise\">\n<div id=\"fs-id1165135154517\" class=\"problem textbox shaded\">\n<h3>Example 11: Finding the <em>y<\/em>&#8211; and <em>x<\/em>-Intercepts of a Polynomial in Factored Form<\/h3>\n<p id=\"fs-id1165135528940\">Find the <i>y<\/i>&#8211;\u00a0and <em>x<\/em>-intercepts of [latex]g\\left(x\\right)={\\left(x - 2\\right)}^{2}\\left(2x+3\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q180029\">Show Solution<\/span><\/p>\n<div id=\"q180029\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135421555\">The <em>y<\/em>-intercept can be found by evaluating [latex]g\\left(0\\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]g\\left(0\\right)={\\left(0 - 2\\right)}^{2}\\left(2\\left(0\\right)+3\\right)=12[\/latex]<\/p>\n<p id=\"eip-id1165134130215\">So the <em>y<\/em>-intercept is [latex]\\left(0,12\\right)[\/latex].<\/p>\n<p id=\"fs-id1165137870836\">The <em>x<\/em>-intercepts can be found by solving [latex]g\\left(x\\right)=0[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]{\\left(x - 2\\right)}^{2}\\left(2x+3\\right)=0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{\\left(x - 2\\right)}^{2}=0 && 2x+3=0 \\\\ &x=2 &&x=-\\frac{3}{2} \\end{align}[\/latex]<\/p>\n<p id=\"eip-id1165135518219\">So the <em>x<\/em>-intercepts are [latex]\\left(2,0\\right)[\/latex] and [latex]\\left(-\\frac{3}{2},0\\right)[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We can always check that our answers are reasonable by using a graphing calculator to graph the polynomial as shown in Figure 11.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"small\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010731\/CNX_Precalc_Figure_03_04_0052.jpg\" alt=\"Graph of f(x)=x^3-5x^2-x+5 with its three intercepts (-1, 0), (1, 0), and (5, 0).\" width=\"487\" height=\"670\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 11<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_04_05\" class=\"example\">\n<div id=\"fs-id1165137415980\" class=\"exercise\">\n<div id=\"fs-id1165134381752\" class=\"problem textbox shaded\">\n<h3>Example 12: Finding the <em>x<\/em>-Intercepts of a Polynomial Function Using a Graph<\/h3>\n<p id=\"fs-id1165137453950\">Find the <em>x<\/em>-intercepts of [latex]h\\left(x\\right)={x}^{3}+4{x}^{2}+x - 6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q512408\">Show Solution<\/span><\/p>\n<div id=\"q512408\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137895270\">This polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques previously discussed. Fortunately, we can use technology to find the intercepts. Keep in mind that some values make graphing difficult by hand. In these cases, we can take advantage of graphing utilities.<\/p>\n<p>Looking at the graph of this function, as shown in Figure 12, it appears that there are <em>x<\/em>-intercepts at [latex]x=-3,-2[\/latex], and 1.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010731\/CNX_Precalc_Figure_03_04_0062.jpg\" alt=\"Graph of g(x)=(x-2)^2(2x+3) with its two x-intercepts (2, 0) and (-3\/2, 0) and its y-intercept (0, 12).\" width=\"487\" height=\"440\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 12<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165131891784\">We can check whether these are correct by substituting these values for <em>x<\/em>\u00a0and verifying that the function is equal to 0.<\/p>\n<p id=\"fs-id1165135600839\">Since [latex]h\\left(x\\right)={x}^{3}+4{x}^{2}+x - 6[\/latex], we have:<\/p>\n<p style=\"text-align: center;\">[latex]h\\left(-3\\right)={\\left(-3\\right)}^{3}+4{\\left(-3\\right)}^{2}+\\left(-3\\right)-6=-27+36 - 3-6=0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]h\\left(-2\\right)={\\left(-2\\right)}^{3}+4{\\left(-2\\right)}^{2}+\\left(-2\\right)-6=-8+16 - 2-6=0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]h\\left(1\\right)={\\left(1\\right)}^{3}+4{\\left(1\\right)}^{2}+\\left(1\\right)-6=1+4+1 - 6=0[\/latex]<\/p>\n<p id=\"fs-id1165134129941\">Each <em>x<\/em>-intercept corresponds to a zero of the polynomial function and each zero yields a factor, so we can now write the polynomial in factored form.<\/p>\n<p style=\"text-align: center;\">[latex]h\\left(x\\right)={x}^{3}+4{x}^{2}+x - 6=\\left(x+3\\right)\\left(x+2\\right)\\left(x - 1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165133238478\">Find the <em>y<\/em>&#8211;\u00a0and <em>x<\/em>-intercepts of the function [latex]f\\left(x\\right)={x}^{4}-19{x}^{2}+30x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q123401\">Show Solution<\/span><\/p>\n<div id=\"q123401\" class=\"hidden-answer\" style=\"display: none\">\n<p>y-intercept [latex]\\left(0,0\\right)[\/latex]; x-intercepts [latex]\\left(0,0\\right),\\left(-5,0\\right),\\left(2,0\\right)[\/latex], and [latex]\\left(3,0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm66678\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=66678&theme=oea&iframe_resize_id=ohm66678\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>\u00a0Determine end behavior<\/h2>\n<p id=\"fs-id1165135514626\">As we have already learned, the behavior of a graph of a <strong>polynomial function<\/strong> of the form<\/p>\n<div id=\"eip-263\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+...+{a}_{1}x+{a}_{0}[\/latex]<\/div>\n<p id=\"eip-id1165134547362\">will either ultimately rise or fall as <em>x<\/em>\u00a0increases without bound and will either rise or fall as <em>x\u00a0<\/em>decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say \u2013100 or \u20131,000.<\/p>\n<p id=\"fs-id1165132959259\">Recall that we call this behavior the <em>end behavior<\/em> of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function, [latex]{a}_{n}{x}^{n}[\/latex], is an even power function, as <em>x<\/em>\u00a0increases or decreases without bound, [latex]f\\left(x\\right)[\/latex] increases without bound. When the leading term is an odd power function, as\u00a0<em>x<\/em>\u00a0decreases without bound, [latex]f\\left(x\\right)[\/latex] also decreases without bound; as <em>x<\/em>\u00a0increases without bound, [latex]f\\left(x\\right)[\/latex] also increases without bound. If the leading term is negative, it will change the direction of the end behavior. The table below\u00a0summarizes all four cases.<\/p>\n<table>\n<thead>\n<tr>\n<th>Even Degree<\/th>\n<th>Odd Degree<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/11.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12504\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/11.png\" alt=\"11\" width=\"423\" height=\"559\" \/><\/a><\/td>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/12.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12505\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/12.png\" alt=\"12\" width=\"397\" height=\"560\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/13.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12506\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/13.png\" alt=\"13\" width=\"387\" height=\"574\" \/><\/a><\/td>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/14.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12507\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/14.png\" alt=\"14\" width=\"404\" height=\"564\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Writing Formulas for Polynomial Functions<\/h2>\n<p id=\"fs-id1165135369122\">Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a <strong>polynomial function<\/strong> written in factored form will have an <em>x<\/em>-intercept where each factor is equal to zero, we can form a function that will pass through a set of <em>x<\/em>-intercepts by introducing a corresponding set of factors.<\/p>\n<div id=\"fs-id1165133320785\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Factored Form of Polynomials<\/h3>\n<p id=\"fs-id1165133320793\">If a polynomial of lowest degree <em>p<\/em>\u00a0has horizontal intercepts at [latex]x={x}_{1},{x}_{2},\\dots ,{x}_{n}[\/latex],\u00a0then the polynomial can be written in the factored form: [latex]f\\left(x\\right)=a{\\left(x-{x}_{1}\\right)}^{{p}_{1}}{\\left(x-{x}_{2}\\right)}^{{p}_{2}}\\cdots {\\left(x-{x}_{n}\\right)}^{{p}_{n}}[\/latex]\u00a0where the powers [latex]{p}_{i}[\/latex]\u00a0on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor <em>a<\/em>\u00a0can be determined given a value of the function other than the <em>x<\/em>-intercept.<\/p>\n<\/div>\n<div id=\"fs-id1165135580289\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135580296\">How To: Given a graph of a polynomial function, write a formula for the function.<\/h3>\n<ol id=\"fs-id1165133309878\">\n<li>Identify the <em>x<\/em>-intercepts of the graph to find the factors of the polynomial.<\/li>\n<li>Examine the behavior of the graph at the <em>x<\/em>-intercepts to determine the multiplicity of each factor.<\/li>\n<li>Find the polynomial of least degree containing all the factors found in the previous step.<\/li>\n<li>Use any other point on the graph (the <em>y<\/em>-intercept may be easiest) to determine the stretch factor.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_04_10\" class=\"example\">\n<div id=\"fs-id1165134043949\" class=\"exercise\">\n<div id=\"fs-id1165134043951\" class=\"problem textbox shaded\">\n<h3>Example 13: Writing a Formula for a Polynomial Function from the Graph<\/h3>\n<p>Write a formula for the polynomial function shown in Figure 13.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010735\/CNX_Precalc_Figure_03_04_0242.jpg\" alt=\"Graph of a positive even-degree polynomial with zeros at x=-3, 2, 5 and y=-2.\" width=\"487\" height=\"366\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 13<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q574656\">Show Solution<\/span><\/p>\n<div id=\"q574656\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135621955\">his graph has three <em>x<\/em>-intercepts: <em>x\u00a0<\/em>= \u20133, 2, and 5. The <em>y<\/em>-intercept is located at (0, 2). At <em>x\u00a0<\/em>= \u20133 and <em>x\u00a0<\/em>= 5,\u00a0the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear. At <em>x\u00a0<\/em>= 2, the graph bounces at the intercept, suggesting the corresponding factor of the polynomial will be second degree (quadratic). Together, this gives us<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a\\left(x+3\\right){\\left(x - 2\\right)}^{2}\\left(x - 5\\right)[\/latex]<\/p>\n<p id=\"fs-id1165135575901\">To determine the stretch factor, we utilize another point on the graph. We will use the <em>y<\/em>-intercept (0, \u20132), to solve for <em>a<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}f\\left(0\\right)&=a\\left(0+3\\right){\\left(0 - 2\\right)}^{2}\\left(0 - 5\\right) \\\\ -2&=a\\left(0+3\\right){\\left(0 - 2\\right)}^{2}\\left(0 - 5\\right) \\\\ -2&=-60a \\\\ a&=\\frac{1}{30} \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165133437286\">The graphed polynomial appears to represent the function [latex]f\\left(x\\right)=\\frac{1}{30}\\left(x+3\\right){\\left(x - 2\\right)}^{2}\\left(x - 5\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Given the graph in Figure 14, write a formula for the function shown.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010735\/CNX_Precalc_Figure_03_04_0252.jpg\" alt=\"Graph of a negative even-degree polynomial with zeros at x=-1, 2, 4 and y=-4.\" width=\"487\" height=\"291\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 14<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q412515\">Show Solution<\/span><\/p>\n<div id=\"q412515\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f\\left(x\\right)=-\\frac{1}{8}{\\left(x - 2\\right)}^{3}{\\left(x+1\\right)}^{2}\\left(x - 4\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135440065\">\n<div class=\"textbox key-takeaways\">\n<h3>Try It 11<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm15942\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15942&theme=oea&iframe_resize_id=ohm15942\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Using Local and Global Extrema<\/h2>\n<p id=\"fs-id1165135440070\">With quadratics, we were able to algebraically find the maximum or minimum value of the function by finding the vertex. For general polynomials, finding these turning points is not possible without more advanced techniques from calculus. Even then, finding where extrema occur can still be algebraically challenging. For now, we will estimate the locations of turning points using technology to generate a graph.<\/p>\n<p id=\"fs-id1165135440077\">Each turning point represents a local minimum or maximum. Sometimes, a turning point is the highest or lowest point on the entire graph. In these cases, we say that the turning point is a <strong>global maximum <\/strong>or a <strong>global minimum<\/strong>. These are also referred to as the absolute maximum and absolute minimum values of the function.<\/p>\n<div id=\"fs-id1165133248530\" class=\"note\">\n<h3 class=\"title\">Local and Global Extrema<\/h3>\n<p id=\"fs-id1165133248538\">A <strong>local maximum<\/strong> or <strong>local minimum<\/strong> at <em>x\u00a0<\/em>= <em>a<\/em>\u00a0(sometimes called the relative maximum or minimum, respectively) is the output at the highest or lowest point on the graph in an open interval around <em>x\u00a0<\/em>= <em>a<\/em>. If a function has a local maximum at <em>a<\/em>, then [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex] for all <em>x<\/em>\u00a0in an open interval around <em>x<\/em> =\u00a0<em>a<\/em>. If a function has a local minimum at <em>a<\/em>, then [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex] for all <em>x<\/em>\u00a0in an open interval around <em>x\u00a0<\/em>= <em>a<\/em>.<\/p>\n<p id=\"fs-id1165134372821\">A <strong>global maximum<\/strong> or <strong>global minimum<\/strong> is the output at the highest or lowest point of the function. If a function has a global maximum at <em>a<\/em>, then [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex] for all <em>x<\/em>. If a function has a global minimum at <em>a<\/em>, then [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex] for all <em>x<\/em>.<\/p>\n<p>We can see the difference between local and global extrema in Figure 15.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010735\/CNX_Precalc_Figure_03_04_026n2.jpg\" alt=\"Graph of an even-degree polynomial that denotes the local maximum and minimum and the global maximum.\" width=\"487\" height=\"475\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 15<\/b><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135347671\" class=\"note precalculus qa textbox\">\n<h3>Q &amp; A<\/h3>\n<p id=\"fs-id1165134422158\"><strong>Do all polynomial functions have a global minimum or maximum?<\/strong><\/p>\n<p id=\"fs-id1165134422162\"><em>No. Only polynomial functions of even degree have a global minimum or maximum. For example, [latex]f\\left(x\\right)=x[\/latex] has neither a global maximum nor a global minimum.<\/em><\/p>\n<\/div>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Key Equations<\/span><\/p>\n<section id=\"fs-id1165137724050\" class=\"key-equations\">\n<table id=\"eip-id1165134063974\" summary=\"..\">\n<tbody>\n<tr>\n<td>general form of a polynomial function<\/td>\n<td>[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165137846272\">\n<li>A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power.<\/li>\n<li>The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.<\/li>\n<li>A polynomial of degree <em>n<\/em>\u00a0will have at most <em>n<\/em>\u00a0<em>x-<\/em>intercepts and at most <em>n<\/em> \u2013 1\u00a0turning points.<\/li>\n<li>The graph of a polynomial function changes direction at its turning points.<\/li>\n<li>Polynomial functions of degree 2 or more are smooth, continuous functions.<\/li>\n<li>To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.<\/li>\n<li>Another way to find the <em>x-<\/em>intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the <em>x<\/em>-axis.<\/li>\n<li>The multiplicity of a zero determines how the graph behaves at the <em>x<\/em>-intercepts, in addition to the behavior equations at each zero.<\/li>\n<li>The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.<\/li>\n<li>The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.<\/li>\n<li>The end behavior of a polynomial function depends on the leading term.<\/li>\n<\/ul>\n<div>\n<section id=\"fs-id1165137731646\" class=\"key-concepts\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137668266\" class=\"definition\">\n<dt><strong>coefficient<\/strong><\/dt>\n<dd id=\"fs-id1165135194915\">a nonzero real number multiplied by a variable raised to an exponent<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135194918\" class=\"definition\">\n<dt><strong>continuous function<\/strong><\/dt>\n<dd id=\"fs-id1165135194921\">a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137832108\" class=\"definition\">\n<dt><strong>degree<\/strong><\/dt>\n<dd id=\"fs-id1165137832112\">the highest power of the variable that occurs in a polynomial<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137832115\" class=\"definition\">\n<dt><strong>end behavior<\/strong><\/dt>\n<dd id=\"fs-id1165131990654\">the behavior of the graph of a function as the input decreases without bound and increases without bound<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135347545\" class=\"definition\">\n<dt><strong>global maximum<\/strong><\/dt>\n<dd id=\"fs-id1165134043812\">highest turning point on a graph; [latex]f\\left(a\\right)[\/latex]\u00a0where [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex]\u00a0for all <em>x<\/em>.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131852045\" class=\"definition\">\n<dt><strong>global minimum<\/strong><\/dt>\n<dd id=\"fs-id1165131852049\">lowest turning point on a graph; [latex]f\\left(a\\right)[\/latex]\u00a0where [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex]<br \/>\nfor all <em>x<\/em>.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131990658\" class=\"definition\">\n<dt><strong>leading coefficient<\/strong><\/dt>\n<dd id=\"fs-id1165131990661\">the coefficient of the leading term<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n<dt><strong>leading term<\/strong><\/dt>\n<dd id=\"fs-id1165132943525\">the term containing the highest power of the variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134112772\" class=\"definition\">\n<dt><strong>multiplicity<\/strong><\/dt>\n<dd id=\"fs-id1165134112776\">the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], [latex]x=h[\/latex]\u00a0is a zero of multiplicity <em>p<\/em>.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943528\" class=\"definition\">\n<dt><strong>polynomial function<\/strong><\/dt>\n<dd id=\"fs-id1165134297639\">a function that 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.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n<dt><strong>term of a polynomial function<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">any [latex]{a}_{i}{x}^{i}[\/latex]\u00a0of a polynomial function in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n<dt><strong>turning point<\/strong><\/dt>\n<dd id=\"fs-id1165133085665\">the location at which the graph of a function changes direction<\/dd>\n<\/dl>\n<\/section>\n<\/div>\n<dl id=\"fs-id1165134112772\" class=\"definition\">\n<dd id=\"fs-id1165134112776\"><\/dd>\n<\/dl>\n<p>&nbsp;<\/p>\n<h2 style=\"text-align: center;\">Section 4.1 Homework Exercises<\/h2>\n<p>1. What is the difference between an <em style=\"font-size: 1rem; font-weight: normal; text-align: initial; color: #373d3f;\">x<\/em><span style=\"font-size: 1rem; font-weight: normal; text-align: initial; color: #373d3f;\">-intercept and a zero of a polynomial function\u00a0[latex]y[\/latex]?<\/span><\/p>\n<p>2.\u00a0If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?<\/p>\n<p>3.\u00a0What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?<\/p>\n<p>4. Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.<\/p>\n<p>5.\u00a0Explain how the factored form of the polynomial helps us in graphing it.<\/p>\n<p>6. If the graph of a polynomial just touches the <em>x<\/em>-axis and then changes direction, what can we conclude about the factored form of the polynomial?<\/p>\n<p>For the following exercises, determine whether the function is polynomial function.\u00a0 If it is a polynomial function, indicate the degree and [latex]a_{n}[\/latex] (leading coefficient).<\/p>\n<p>7. [latex]\\text{ }f\\left(x\\right)=6{x}^{5}[\/latex]<\/p>\n<p>8. [latex]\\text{ }f\\left(x\\right)={\\left({x}^{2}\\right)}^{3}[\/latex]<\/p>\n<p>9. [latex]\\text{ }f\\left(x\\right)=\\frac{{x}^{2}}{{x}^{2}-1}[\/latex]<\/p>\n<p>10.\u00a0[latex]\\text{ }f\\left(x\\right)=x-{x}^{4}[\/latex]<\/p>\n<p>11.\u00a0[latex]\\text{ }f\\left(x\\right)=2x\\left(x+2\\right){\\left(x - 1\\right)}^{2}[\/latex]<\/p>\n<p>12. [latex]\\text{ }f\\left(x\\right)={3}^{x+1}[\/latex]<\/p>\n<p>13. [latex]\\text{ }f\\left(x\\right)=5\\sqrt(x)+2x^{4}[\/latex]<\/p>\n<p>14. [latex]\\text{ }f\\left(x\\right)=4x^{2}-\\dfrac{3}{x^{3}}[\/latex]<\/p>\n<p>For the following exercises, find the intercepts of the functions.<\/p>\n<p>15. [latex]f\\left(t\\right)=2\\left(t - 1\\right)\\left(t+2\\right)\\left(t - 3\\right)[\/latex]<\/p>\n<p>16.\u00a0[latex]g\\left(n\\right)=-2\\left(3n - 1\\right)\\left(2n+1\\right)[\/latex]<\/p>\n<p>17.\u00a0[latex]f\\left(x\\right)=-4(x+2)(x-2)[\/latex]<\/p>\n<p>18.\u00a0[latex]f\\left(x\\right)=\\left(x+3\\right)\\left(4{x}^{2}-1\\right)[\/latex]<\/p>\n<p>19. [latex]f\\left(x\\right)=x\\left({x}^{2}-2x - 8\\right)[\/latex]<\/p>\n<p>For the following exercises, determine the least possible degree of the polynomial function shown.<\/p>\n<p>20.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005215\/CNX_Precalc_Figure_03_03_201.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/><\/p>\n<p>21.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005216\/CNX_Precalc_Figure_03_03_202.jpg\" alt=\"Graph of an even-degree polynomial.\" \/><\/p>\n<p>22.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005216\/CNX_Precalc_Figure_03_03_203.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/><\/p>\n<p>23.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005216\/CNX_Precalc_Figure_03_03_204.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/><\/p>\n<p>24.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005216\/CNX_Precalc_Figure_03_03_205.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/><\/p>\n<p>25.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005216\/CNX_Precalc_Figure_03_03_206.jpg\" alt=\"Graph of an even-degree polynomial.\" \/><\/p>\n<p>26.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005217\/CNX_Precalc_Figure_03_03_207.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/><\/p>\n<p>For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.<\/p>\n<p>27.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005217\/CNX_Precalc_Figure_03_03_209.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/><\/p>\n<p>28.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005219\/CNX_Precalc_Figure_03_03_210.jpg\" alt=\"Graph of an equation.\" \/><\/p>\n<p>29.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005220\/CNX_Precalc_Figure_03_03_211.jpg\" alt=\"Graph of an even-degree polynomial.\" \/><\/p>\n<p>30.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005220\/CNX_Precalc_Figure_03_03_212.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/><\/p>\n<p>31.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005221\/CNX_Precalc_Figure_03_03_213.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/><\/p>\n<p>32.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005221\/CNX_Precalc_Figure_03_03_214.jpg\" alt=\"Graph of an equation.\" \/><\/p>\n<p>33.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005221\/CNX_Precalc_Figure_03_03_215.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/><\/p>\n<p>For the following exercises, find the zeros and multiplicities.<\/p>\n<p>34. [latex]f\\left(x\\right)={\\left(x+2\\right)}^{3}{\\left(x - 3\\right)}^{2}[\/latex]<\/p>\n<p>35. [latex]f\\left(x\\right)={x}^{2}{\\left(2x+3\\right)}^{3}{\\left(x - 4\\right)}^{2}[\/latex]<\/p>\n<p>36.\u00a0[latex]f\\left(x\\right)={x}^{3}{\\left(x - 1\\right)}^{3}\\left(x+2\\right)[\/latex]<\/p>\n<p>37. [latex]f\\left(x\\right)={x}^{2}\\left({x}^{2}+4x+4\\right)[\/latex]<\/p>\n<p>38.\u00a0[latex]f\\left(x\\right)={\\left(2x+1\\right)}^{3}\\left(9{x}^{2}-6x+1\\right)[\/latex]<\/p>\n<p>39. [latex]f\\left(x\\right)={\\left(3x+2\\right)}^{3}\\left({x}^{2}-10x+25\\right)[\/latex]<\/p>\n<p>40.\u00a0[latex]f\\left(x\\right)=x\\left(4{x}^{2}-12x+9\\right)\\left({x}^{2}+8x+16\\right)[\/latex]<\/p>\n<p>41. [latex]f\\left(x\\right)={x}^{6}-{x}^{5}-2{x}^{4}[\/latex]<\/p>\n<p>32.\u00a0[latex]f\\left(x\\right)=3{x}^{4}+6{x}^{3}+3{x}^{2}[\/latex]<\/p>\n<p>43. [latex]f\\left(x\\right)=4{x}^{5}-12{x}^{4}+9{x}^{3}[\/latex]<\/p>\n<p>44.\u00a0[latex]f\\left(x\\right)=2{x}^{4}\\left({x}^{3}-4{x}^{2}+4x\\right)[\/latex]<\/p>\n<p>45. [latex]f\\left(x\\right)=4{x}^{4}\\left(9{x}^{4}-12{x}^{3}+4{x}^{2}\\right)[\/latex]<\/p>\n<p>For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or \u20131. There may be more than one correct answer.<\/p>\n<p>46. The <em>y<\/em>-intercept is [latex]\\left(0,-6\\right)[\/latex]. The <em>x<\/em>-intercepts are [latex]\\left(-3,0\\right),\\left(2,0\\right)[\/latex]. Degree is 2.<\/p>\n<p>End behavior: [latex]\\text{as }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex].<\/p>\n<p>47. The <em>y<\/em>-intercept is [latex]\\left(0,-4\\right)[\/latex]. The <em>x<\/em>-intercepts are [latex]\\left(-2,0\\right),\\left(2,0\\right)[\/latex]. Degree is 2.<\/p>\n<p>End behavior: [latex]\\text{as }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex].<\/p>\n<p>48.\u00a0The <em>y<\/em>-intercept is [latex]\\left(0,9\\right)[\/latex]. The <em>x<\/em>-intercepts are [latex]\\left(-3,0\\right),\\left(3,0\\right)[\/latex]. Degree is 2.<\/p>\n<p>End behavior: [latex]\\text{as }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to -\\infty[\/latex].<\/p>\n<p>49. The <em>y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex]. The <em>x<\/em>-intercepts are [latex]\\left(0,0\\right),\\left(2,0\\right)[\/latex]. Degree is 3.<\/p>\n<p>End behavior: [latex]\\text{as }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex].<\/p>\n<p>50.\u00a0The <em>y-<\/em>intercept is [latex]\\left(0,1\\right)[\/latex]. The <em>x<\/em>-intercept is [latex]\\left(1,0\\right)[\/latex]. Degree is 3.<\/p>\n<p>End behavior: [latex]\\text{as }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to -\\infty[\/latex].<\/p>\n<p>51. The <em>y<\/em>-intercept is [latex]\\left(0,1\\right)[\/latex]. There is no <em>x<\/em>-intercept. Degree is 4.<\/p>\n<p>End behavior: [latex]\\text{as }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex].<\/p>\n<p>For the following exercises, use the graphs to write the formula for a polynomial function of least degree.<\/p>\n<p>52.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005231\/CNX_Precalc_Figure_03_04_207.jpg\" alt=\"Graph of a positive odd-degree polynomial with zeros at x=-2, 1, and 3.\" \/><\/p>\n<p>53.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005231\/CNX_PreCalc_Figure_03_04_208.jpg\" alt=\"Graph of a negative odd-degree polynomial with zeros at x=-3, 1, and 3.\" \/><br \/>\n54.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005232\/CNX_PreCalc_Figure_03_04_209.jpg\" alt=\"Graph of a negative odd-degree polynomial with zeros at x=-1, and 2.\" \/><\/p>\n<p>55.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005232\/CNX_PreCalc_Figure_03_04_210.jpg\" alt=\"Graph of a positive odd-degree polynomial with zeros at x=-2, and 3.\" \/><\/p>\n<p>56.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005232\/CNX_PreCalc_Figure_03_04_211.jpg\" alt=\"Graph of a negative even-degree polynomial with zeros at x=-3, -2, 3, and 4.\" \/><\/p>\n<p>57.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005232\/CNX_PreCalc_Figure_03_04_212.jpg\" alt=\"Graph of a negative even-degree polynomial with zeros at x=-4, -2, 1, and 3.\" \/><br \/>\n58.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005233\/CNX_PreCalc_Figure_03_04_213.jpg\" alt=\"Graph of a positive even-degree polynomial with zeros at x=-4, -2, and 3.\" \/><\/p>\n<p>59.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005233\/CNX_PreCalc_Figure_03_04_215.jpg\" alt=\"Graph of a negative odd-degree polynomial with zeros at x=-3, -2, and 1.\" \/><\/p>\n<\/section>\n<\/section>\n","protected":false},"author":264444,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-16686","chapter","type-chapter","status-publish","hentry"],"part":17771,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/16686","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":72,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/16686\/revisions"}],"predecessor-version":[{"id":17944,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/16686\/revisions\/17944"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/parts\/17771"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/16686\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=16686"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=16686"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=16686"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=16686"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}