{"id":13852,"date":"2018-08-24T22:13:20","date_gmt":"2018-08-24T22:13:20","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalcone\/?post_type=chapter&#038;p=13852"},"modified":"2025-02-05T05:19:15","modified_gmt":"2025-02-05T05:19:15","slug":"graphs-of-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/graphs-of-polynomial-functions\/","title":{"raw":"Graphs of Polynomial Functions","rendered":"Graphs of Polynomial Functions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Recognize characteristics of graphs of polynomial functions.<\/li>\r\n \t<li style=\"font-weight: 400;\">Identify zeros of polynomials and their multiplicities.<\/li>\r\n \t<li style=\"font-weight: 400;\">Determine end behavior.<\/li>\r\n \t<li style=\"font-weight: 400;\">Understand the relationship between degree and turning points.<\/li>\r\n \t<li style=\"font-weight: 400;\">Graph polynomial functions.<\/li>\r\n \t<li style=\"font-weight: 400;\">Solve polynomial inequalities.<\/li>\r\n \t<li style=\"font-weight: 400;\">Use the Intermediate Value Theorem.<\/li>\r\n \t<li>Write the formula for a polynomial function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165135545777\">The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in the table below<b>.<\/b><\/p>\r\n\r\n<table id=\"Table_03_04_01\" summary=\"Two rows and nine columns. The first row is labeled, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>Year<\/strong><\/td>\r\n<td>2006<\/td>\r\n<td>2007<\/td>\r\n<td>2008<\/td>\r\n<td>2009<\/td>\r\n<td>2010<\/td>\r\n<td>2011<\/td>\r\n<td>2012<\/td>\r\n<td>2013<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Revenues<\/strong><\/td>\r\n<td>52.4<\/td>\r\n<td>52.8<\/td>\r\n<td>51.2<\/td>\r\n<td>49.5<\/td>\r\n<td>48.6<\/td>\r\n<td>48.6<\/td>\r\n<td>48.7<\/td>\r\n<td>47.1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165134040487\">The revenue can be modeled by the polynomial function<\/p>\r\n\r\n<div id=\"eip-679\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]R\\left(t\\right)=-0.037{t}^{4}+1.414{t}^{3}-19.777{t}^{2}+118.696t - 205.332[\/latex]<\/div>\r\n<p id=\"fs-id1165137659450\">where <em>R<\/em>\u00a0represents the revenue in millions of dollars and <em>t<\/em>\u00a0represents the year, with <em>t<\/em> = 6\u00a0corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.<\/p>\r\n\r\n<h2>Recognize characteristics of graphs of polynomial functions<\/h2>\r\n<p id=\"fs-id1165134352567\">Polynomial functions of degree 2 or more have graphs that do not have sharp corners; recall that these types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. Figure 1 shows\u00a0a graph that represents a <strong>polynomial function<\/strong> and a graph that represents a function that is not a polynomial.<span id=\"fs-id1165135185916\">\r\n<\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"900\"]<img class=\"\" 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_0012.jpg\" alt=\"Graph of f(x)=x^3-0.01x.\" width=\"900\" height=\"409\" \/> <b>Figure 1<\/b>[\/caption]\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 1: Recognizing Polynomial Functions<\/h3>\r\nWhich of the graphs in Figure 2\u00a0represents a 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 2<\/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-id1165134129608\">The graphs of <em>f<\/em>\u00a0and <em>h<\/em>\u00a0are graphs of polynomial functions. They are smooth and <strong>continuous<\/strong>.<\/p>\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\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134164967\" class=\"note precalculus qa textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<p id=\"fs-id1165135496631\"><strong>Do all polynomial functions have as their domain all real numbers?<\/strong><\/p>\r\n<p id=\"fs-id1165134342693\"><em>Yes. Any real number is a valid input for a polynomial function.<\/em><\/p>\r\n\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 2: 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 3<\/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 3: 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 4<\/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 4: 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 5.\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 5<\/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 5: 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 6, 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 6<\/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 2<\/h3>\r\n[ohm_question hide_question_numbers=1]66678[\/ohm_question]\r\n\r\n<\/div>\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 7\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 7.<\/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 8\u00a0for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.\r\n\r\n[caption id=\"attachment_16092\" align=\"aligncenter\" width=\"874\"]<a href=\"https:\/\/courses.lumenlearning.com\/precalculus\/wp-content\/uploads\/sites\/3675\/2018\/08\/404d5117e8c2b2cc187c001d0fcf267e8d3c7bbf.jpeg\"><img class=\"wp-image-16092 size-full\" src=\"https:\/\/courses.lumenlearning.com\/precalculus\/wp-content\/uploads\/sites\/3675\/2018\/08\/404d5117e8c2b2cc187c001d0fcf267e8d3c7bbf.jpeg\" alt=\"Three graphs, left to right, with zeros of multiplicity 1, 2, and 3.\" width=\"874\" height=\"324\" \/><\/a> Figure 8.[\/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 6: 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 9<\/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 9 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 10<\/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 3, a zero of \u20131 with multiplicity 2, and a zero of 3 with even multiplicity of 4.\r\n\r\n[\/hidden-answer]\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>Understand the relationship between degree and turning points<\/h2>\r\n<p id=\"fs-id1165135416524\">In addition to the end behavior, recall that we can analyze a polynomial function\u2019s local behavior. It may have a turning point where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). Look at the graph of the polynomial function [latex]f\\left(x\\right)={x}^{4}-{x}^{3}-4{x}^{2}+4x[\/latex] in Figure 11. 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 11<\/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 7: 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<h2>\u00a0Graph polynomial functions<\/h2>\r\n<p id=\"fs-id1165137843095\">We can use what we have learned about multiplicities, end behavior, and turning points to sketch graphs of polynomial functions. Let us put this all together and look at the steps required to graph polynomial functions.<\/p>\r\n\r\n<div id=\"fs-id1165137843101\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135449677\">How To: Given a polynomial function, sketch the graph.<\/h3>\r\n<ol id=\"fs-id1165135449683\">\r\n \t<li>Find the intercepts.<\/li>\r\n \t<li>Check for symmetry. If the function is an even function, its graph is symmetrical about the <em>y<\/em>-axis, that is,\u00a0<em>f<\/em>(\u2013<em>x<\/em>) = <em>f<\/em>(<em>x<\/em>).\r\nIf a function is an odd function, its graph is symmetrical about the origin, that is,\u00a0<em>f<\/em>(\u2013<em>x<\/em>) = <em>\u2013<\/em><em>f<\/em>(<em>x<\/em>).<\/li>\r\n \t<li>Use the multiplicities of the zeros to determine the behavior of the polynomial at the <em>x<\/em>-intercepts.<\/li>\r\n \t<li>Determine the end behavior by examining the leading term.<\/li>\r\n \t<li>Use the end behavior and the behavior at the intercepts to sketch a graph.<\/li>\r\n \t<li>Ensure that the number of turning points does not exceed one less than the degree of the polynomial.<\/li>\r\n \t<li>Optionally, use technology to check the graph.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_04_08\" class=\"example\">\r\n<div id=\"fs-id1165135575951\" class=\"exercise\">\r\n<div id=\"fs-id1165135575953\" class=\"problem textbox shaded\">\r\n<h3>Example 8: Sketching the Graph of a Polynomial Function<\/h3>\r\n<p id=\"fs-id1165135575958\">Sketch a graph of [latex]f\\left(x\\right)=-2{\\left(x+3\\right)}^{2}\\left(x - 5\\right)[\/latex].<\/p>\r\n[reveal-answer q=\"892446\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"892446\"]\r\n<p id=\"fs-id1165135237929\">This graph has two <em>x-<\/em>intercepts. At <em>x\u00a0<\/em>= \u20133, the factor is squared, indicating a multiplicity of 2. The graph will bounce at this <em>x<\/em>-intercept. At <em>x\u00a0<\/em>= 5, the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept.<\/p>\r\n<p id=\"fs-id1165135171021\">The <em>y<\/em>-intercept is found by evaluating <em>f<\/em>(0).<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align} f\\left(0\\right)&amp;=-2{\\left(0+3\\right)}^{2}\\left(0 - 5\\right) \\\\ &amp;=-2\\cdot 9\\cdot \\left(-5\\right) \\\\ &amp;=90 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165134374772\">The <em>y<\/em>-intercept is (0, 90).<\/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_0172.jpg\" alt=\"Showing the distribution for the leading term.\" width=\"487\" height=\"362\" \/> <b>Figure 13<\/b>[\/caption]\r\n<p id=\"fs-id1165134381522\">Additionally, we can see the leading term, if this polynomial were multiplied out, would be [latex]-2{x}^{3}[\/latex],\r\nso the end behavior is that of a vertically reflected cubic, with the outputs decreasing as the inputs approach infinity, and the outputs increasing as the inputs approach negative infinity.<span id=\"fs-id1165135646080\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165134374738\">To sketch this, we consider that:<\/p>\r\n\r\n<ul id=\"fs-id1165134374741\">\r\n \t<li>As [latex]x\\to -\\infty [\/latex] the function [latex]f\\left(x\\right)\\to \\infty [\/latex], so we know the graph starts in the second quadrant and is decreasing toward the <em>x<\/em>-axis.<\/li>\r\n \t<li>Since [latex]f\\left(-x\\right)=-2{\\left(-x+3\\right)}^{2}\\left(-x - 5\\right)[\/latex]\r\nis not equal to <em>f<\/em>(<em>x<\/em>), the graph does not display symmetry.<\/li>\r\n \t<li>At (-3,0), the graph bounces off of the <em>x<\/em>-axis, so the function must start increasing.\r\n<p id=\"fs-id1165135536183\" style=\"text-align: left;\">At (0, 90), the graph crosses the <em>y<\/em>-axis at the <em>y<\/em>-intercept.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<figure id=\"Figure_03_04_018\" class=\"small\">\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_0182.jpg\" alt=\"Graph of the end behavior and intercepts, (-3, 0) and (0, 90), for the function f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"362\" \/> <b>Figure 14<\/b>[\/caption]<\/figure>\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\/03010734\/CNX_Precalc_Figure_03_04_0192.jpg\" alt=\"Graph of the end behavior and intercepts, (-3, 0), (0, 90) and (5, 0), for the function f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"362\" \/> <b>Figure 15<\/b>[\/caption]\r\n<p id=\"fs-id1165135241000\">Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the graph passes through the next intercept at (5, 0).\u00a0<span id=\"fs-id1165135241013\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165135613608\">As [latex]x\\to \\infty [\/latex] the function [latex]f\\left(x\\right)\\to \\mathrm{-\\infty }[\/latex], so we know the graph continues to decrease, and we can stop drawing the graph in the fourth quadrant.<\/p>\r\n<p id=\"fs-id1165135574296\">Using technology, we can create the graph for the polynomial function, shown in Figure 16, and verify that the resulting graph looks like our sketch in Figure 15.<\/p>\r\n\r\n<figure id=\"Figure_03_04_020\" class=\"small\"><figcaption>The complete graph of the polynomial function [latex]f\\left(x\\right)=-2{\\left(x+3\\right)}^{2}\\left(x - 5\\right)[\/latex]<\/figcaption>\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\/03010734\/CNX_Precalc_Figure_03_04_0202.jpg\" alt=\"Graph of f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"366\" \/> <b>Figure 16<\/b>[\/caption]<\/figure>\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-id1165133065140\">Sketch a graph of [latex]f\\left(x\\right)=\\frac{1}{4}x{\\left(x - 1\\right)}^{4}{\\left(x+3\\right)}^{3}[\/latex].<\/p>\r\n[reveal-answer q=\"408253\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"408253\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010734\/CNX_Precalc_Figure_03_04_0212.jpg\" alt=\"Graph of f(x)=(1\/4)x(x-1)^4(x+3)^3.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>\u00a0Solving Polynomial Inequalities<\/h2>\r\nOne application of our ability to find intercepts and sketch a graph of polynomials is the ability to solve polynomial inequalities. It is a very common question to ask when a function will be positive and negative. We can solve polynomial inequalities by either utilizing the graph, or by using test values.\r\n<div id=\"fs-id1165137433651\" class=\"solution textbox shaded\">\r\n<h3>Example 9: Solving Polynomial Inequalities in Factored From<\/h3>\r\nSolve [latex]\\left(x+3\\right){\\left(x+1\\right)}^{2}\\left(x-4\\right)&gt; 0[\/latex]\r\n\r\n[reveal-answer q=\"494744\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"494744\"]\r\n\r\nAs with all inequalities, we start by solving the equality [latex]\\left(x+3\\right){\\left(x+1\\right)}^{2}\\left(x-4\\right)= 0[\/latex], which has solutions at x = -3, -1, and 4. We know the function can only change from positive to negative at these values, so these divide the inputs into 4 intervals.\r\nWe could choose a test value in each interval and evaluate the function [latex]f\\left(x\\right) = \\left(x+3\\right){\\left(x+1\\right)}^{2}\\left(x-4\\right)[\/latex] at each test value to determine if the function is positive or negative in that interval\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Interval<\/td>\r\n<td>Test x in interval<\/td>\r\n<td>f(test value)<\/td>\r\n<td>&gt; 0 or &lt; 0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>x &lt; -3<\/td>\r\n<td>-4<\/td>\r\n<td>72<\/td>\r\n<td>&gt; 0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>-3 &lt; x &lt; -1<\/td>\r\n<td>-2<\/td>\r\n<td>-6<\/td>\r\n<td>&lt; 0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>-1 &lt;\u00a0 x &lt; 4<\/td>\r\n<td>0<\/td>\r\n<td>-12<\/td>\r\n<td>&lt; 0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>x &gt; 4<\/td>\r\n<td>5<\/td>\r\n<td>288<\/td>\r\n<td>\u00a0&gt; 0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nOn a number line this would look like:\r\n\r\n<img class=\"aligncenter wp-image-13403 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/139\/2016\/04\/28183410\/1.png\" alt=\"Number line with values from -6 to 6 double headed arrows from -6 to -3 read positive, from -3 to -1 read negative, from -1 to positive 4 read negative and from 4 to 6 read positive.\" width=\"630\" height=\"110\" \/>\r\nFrom our test values, we can determine this function is positive when <em>x<\/em> &lt; -3 or <em>x<\/em> &gt; 4, or in interval notation, [latex]\\left(-\\infty, -3\\right)\\cup\\left(4,\\infty\\right)[\/latex]. We could have also determined on which intervals the function was positive by sketching a graph of the function. We illustrate that technique in the next example.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137433651\" class=\"solution textbox shaded\">\r\n<h3>Example 10: Solving Polynomial Inequalities in Factored From<\/h3>\r\nFind the domain of the function [latex]v\\left(t\\right)=\\sqrt{6-5t-{t}^{2}}[\/latex]\r\n\r\n[reveal-answer q=\"359527\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"359527\"]\r\n\r\nA square root is only defined when the quantity we are taking the square root of, the quantity inside the square root, is zero or greater. Thus, the domain of this function will be when [latex]6 - 5t - {t}^{2}\\ge 0[\/latex]. Again we start by solving the equality [latex]6 - 5t - {t}^{2}= 0[\/latex]. While we could use the quadratic formula, this equation factors nicely to [latex]\\left(6 + t\\right)\\left(1-t\\right)=0[\/latex], giving horizontal intercepts\r\nt = 1 and t = -6.\r\nSketching a graph of this quadratic will allow us to determine when it is positive.\r\n\r\n<img class=\"aligncenter size-full wp-image-13404\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/139\/2016\/04\/28183442\/Screen-Shot2.png\" alt=\"Graph of upside down parabola on cartesian coordinate axes passing through (-6,0) and (1,0)\" width=\"278\" height=\"204\" \/>\r\nFrom the graph we can see this function is positive for inputs between the intercepts. So [latex]6 - 5t - {t}^{2}\\ge 0[\/latex] is positive for [latex]-6 \\le t\\le 1[\/latex], and this will be the domain of the v(t) function.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137433651\" class=\"solution textbox shaded\">\r\n<h3>Example 11: Solving a Polynomial Inequality Not in Factored Form<\/h3>\r\nSolve the inequality [latex]{x}^{4} - 2{x}^{3} - 3{x}^{2} \\gt 0[\/latex]\r\n\r\n[reveal-answer q=\"273617\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"273617\"]\r\n\r\nIn our other examples, we were given polynomials that were already in factored form, here we have an additional step to finding the intervals on which solutions to the given inequality lie. Again, we will start by solving the equality [latex]{x}^{4} - 2{x}^{3} - 3{x}^{2} = 0[\/latex]\r\n\r\n&nbsp;\r\n<p style=\"text-align: left;\">Notice that there is a common factor of [latex]{x}^{2}[\/latex] in each term of this polynomial. We can use factoring to simplify in the following way:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}{x}^{4} - 2{x}^{3} - 3{x}^{2} &amp;= 0&amp;\\\\{x}^{2}\\left({x}^{2} - 2{x} - 3\\right) &amp;= 0\\\\ {x}^{2}\\left(x - 3\\right)\\left(x + 1 \\right)&amp;= 0\\end{align}[\/latex]<\/p>\r\nNow we can set each factor equal to zero to find the solution to the equality.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc} {x}^{2} = 0 &amp; \\left(x - 3\\right) = 0 &amp;\\left(x+1\\right) = 0\\\\ {x} = 0 &amp; x = 3 &amp; x = -1\\\\ \\end{array}[\/latex].<\/p>\r\nNote that x = 0 has multiplicity of two, but since our inequality is strictly greater than, we don't need to include it in our solutions.\r\nWe can choose a test value in each interval and evaluate the function\r\n<p style=\"text-align: center;\">[latex]{x}^{4} - 2{x}^{3} - 3{x}^{2} = 0[\/latex]<\/p>\r\n<p style=\"text-align: left;\">at each test value to determine if the function is positive or negative in that interval<\/p>\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Interval<\/td>\r\n<td>Test x in interval<\/td>\r\n<td>&gt; 0,\u00a0 &lt; 0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>x &lt; -1<\/td>\r\n<td>-2<\/td>\r\n<td>x &gt; 0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>-1 &lt; x &lt; 0<\/td>\r\n<td>-1\/2<\/td>\r\n<td>\u00a0x &lt;\u00a0 0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0 &lt; x &lt; 3<\/td>\r\n<td>1<\/td>\r\n<td>x &lt; 0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>x &gt; 3<\/td>\r\n<td>5<\/td>\r\n<td>x &gt; 0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe want to have the set of x values that will give us the intervals where the polynomial is greater than zero. Our answer will be [latex]\\left(-\\infty, -1\\right]\\cup\\left[3,\\infty\\right)[\/latex].\r\n\r\n&nbsp;\r\n\r\nThe graph of the function gives us additional confirmation of our solution.\r\n\r\n<img class=\"aligncenter wp-image-13406 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/139\/2016\/04\/28183505\/Screen-Shot3.png\" alt=\"Line dips down, dips slightly up, dips very far down, then sharply goes up\" width=\"302\" height=\"445\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it 5<\/h3>\r\n[ohm_question]34324[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Further Examples<\/h2>\r\n<h3>Solving a polynomial inequality not in factored form - use factoring by grouping.<\/h3>\r\nhttps:\/\/youtu.be\/jmeLkQCFLHs\r\n<h3>Solving a polynomial inequality not in factored form - use greatest common factor.<\/h3>\r\nhttps:\/\/youtu.be\/zyiad-T6-TI\r\n<h3>Solving a polynomial inequality not in factored form - factor a trinomial<\/h3>\r\nhttps:\/\/youtu.be\/LC1bwRHcdh4\r\n<h2>Use the Intermediate Value Theorem<\/h2>\r\n<p id=\"fs-id1165135205093\">In some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the <em>x<\/em>-axis, we can confirm that there is a zero between them. Consider a polynomial function <em>f<\/em>\u00a0whose graph is smooth and continuous. The <strong>Intermediate Value Theorem<\/strong> states that for two numbers <em>a<\/em>\u00a0and <em>b<\/em>\u00a0in the domain of <em>f<\/em>,\u00a0if <em>a\u00a0<\/em>&lt; <em>b<\/em>\u00a0and [latex]f\\left(a\\right)\\ne f\\left(b\\right)[\/latex], then the function <em>f<\/em>\u00a0takes on every value between [latex]f\\left(a\\right)[\/latex] and [latex]f\\left(b\\right)[\/latex].<\/p>\r\nWe can apply this theorem to a special case that is useful in graphing polynomial functions. If a point on the graph of a continuous function <em>f<\/em>\u00a0at [latex]x=a[\/latex] lies above the <em>x<\/em>-axis and another point at [latex]x=b[\/latex] lies below the <em>x<\/em>-axis, there must exist a third point between [latex]x=a[\/latex] and [latex]x=b[\/latex] where the graph crosses the <em>x<\/em>-axis. Call this point [latex]\\left(c,\\text{ }f\\left(c\\right)\\right)[\/latex]. This means that we are assured there is a solution <em>c<\/em>\u00a0where [latex]f\\left(c\\right)=0[\/latex].\r\n\r\nIn other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the <em>x<\/em>-axis. Figure 17\u00a0shows that there is a zero between <em>a<\/em>\u00a0and <em>b<\/em>.\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\/03010734\/CNX_Precalc_Figure_03_04_0222.jpg\" alt=\"Graph of an odd-degree polynomial function that shows a point f(a) that\u2019s negative, f(b) that\u2019s positive, and f(c) that\u2019s 0.\" width=\"487\" height=\"368\" \/> <b>Figure 17.<\/b> Using the Intermediate Value Theorem to show there exists a zero.[\/caption]\r\n\r\n<div id=\"fs-id1165135347510\" class=\"note textbox shaded\">\r\n<h3 class=\"title\">A General Note: Intermediate Value Theorem<\/h3>\r\n<p id=\"fs-id1165135580347\">Let <em>f<\/em>\u00a0be a polynomial function. The <strong>Intermediate Value Theorem<\/strong> states that if [latex]f\\left(a\\right)[\/latex]\u00a0and [latex]f\\left(b\\right)[\/latex]\u00a0have opposite signs, then there exists at least one value <em>c<\/em>\u00a0between <em>a<\/em>\u00a0and <em>b<\/em>\u00a0for which [latex]f\\left(c\\right)=0[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"Example_03_04_09\" class=\"example\">\r\n<div id=\"fs-id1165133358799\" class=\"exercise\">\r\n<div id=\"fs-id1165133358801\" class=\"problem textbox shaded\">\r\n<h3>Example 12: Using the Intermediate Value Theorem<\/h3>\r\n<p id=\"fs-id1165133358807\">Show that the function [latex]f\\left(x\\right)={x}^{3}-5{x}^{2}+3x+6[\/latex]\u00a0has at least two real zeros between [latex]x=1[\/latex]\u00a0and [latex]x=4[\/latex].<\/p>\r\n[reveal-answer q=\"98939\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"98939\"]\r\n<p id=\"fs-id1165135537349\">As a start, evaluate [latex]f\\left(x\\right)[\/latex]\u00a0at the integer values [latex]x=1,2,3,\\text{ and }4[\/latex].<\/p>\r\n\r\n<table id=\"Table_03_04_03\" summary=\"..\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em><strong>f\u00a0<\/strong><\/em><strong>(<em>x<\/em>)<\/strong><\/td>\r\n<td>5<\/td>\r\n<td>0<\/td>\r\n<td>\u20133<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165135536378\">We see that one zero occurs at [latex]x=2[\/latex]. Also, since [latex]f\\left(3\\right)[\/latex] is negative and [latex]f\\left(4\\right)[\/latex] is positive, by the Intermediate Value Theorem, there must be at least one real zero between 3 and 4.<\/p>\r\n<p id=\"fs-id1165135575934\">We have shown that there are at least two real zeros between [latex]x=1[\/latex]\u00a0and [latex]x=4[\/latex].<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nWe can also see in Figure 18\u00a0that there are two real zeros between [latex]x=1[\/latex]\u00a0and [latex]x=4[\/latex].\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_0232.jpg\" alt=\"Graph of f(x)=x^3-5x^2+3x+6 and shows, by the Intermediate Value Theorem, that there exists two zeros since f(1)=5 and f(4)=2 are positive and f(3) = -3 is negative.\" width=\"487\" height=\"591\" \/> <b>Figure 18<\/b>[\/caption]\r\n\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-id1165135551168\">Show that the function [latex]f\\left(x\\right)=7{x}^{5}-9{x}^{4}-{x}^{2}[\/latex] has at least one real zero between [latex]x=1[\/latex] and [latex]x=2[\/latex].<\/p>\r\n[reveal-answer q=\"886741\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"886741\"]\r\n\r\nBecause <em>f<\/em>\u00a0is a polynomial function and since [latex]f\\left(1\\right)[\/latex] is negative and [latex]f\\left(2\\right)[\/latex] is positive, there is at least one real zero between [latex]x=1[\/latex] and [latex]x=2[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<section id=\"fs-id1165135369116\">\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 19.\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 19<\/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 20, 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 20<\/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<\/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 21.\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 21<\/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<div id=\"Example_03_04_11\" class=\"example\">\r\n<div id=\"fs-id1165135470044\" class=\"exercise\">\r\n<div id=\"fs-id1165135470046\" class=\"problem textbox shaded\">\r\n<h3>Example 14: Using Local Extrema to Solve Applications<\/h3>\r\n<p id=\"fs-id1165135470052\">An open-top box is to be constructed by cutting out squares from each corner of a 14 cm by 20 cm sheet of plastic then folding up the sides. Find the size of squares that should be cut out to maximize the volume enclosed by the box.<\/p>\r\n[reveal-answer q=\"850359\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"850359\"]\r\n\r\nWe will start this problem by drawing a picture like Figure 22, labeling the width of the cut-out squares with a variable, <em>w<\/em>.\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\/03010736\/CNX_Precalc_Figure_03_04_0272.jpg\" alt=\"Diagram of a rectangle with four squares at the corners.\" width=\"487\" height=\"298\" \/> <b>Figure 22<\/b>[\/caption]\r\n<p id=\"fs-id1165134150016\">Notice that after a square is cut out from each end, it leaves a [latex]\\left(14 - 2w\\right)[\/latex] cm by [latex]\\left(20 - 2w\\right)[\/latex] cm rectangle for the base of the box, and the box will be <em>w<\/em>\u00a0cm tall. This gives the volume<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}V\\left(w\\right)&amp;=\\left(20 - 2w\\right)\\left(14 - 2w\\right)w \\\\ &amp;=280w - 68{w}^{2}+4{w}^{3} \\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\/03010736\/CNX_Precalc_Figure_03_04_0282.jpg\" alt=\"Graph of V(w)=(20-2w)(14-2w)w where the x-axis is labeled w and the y-axis is labeled V(w).\" width=\"487\" height=\"406\" \/> <b>Figure 23<\/b>[\/caption]\r\n<p id=\"fs-id1165135628578\">Notice, since the factors are <em>w<\/em>, [latex]20 - 2w[\/latex] and [latex]14 - 2w[\/latex], the three zeros are 10, 7, and 0, respectively. Because a height of 0 cm is not reasonable, we consider the only the zeros 10 and 7. The shortest side is 14 and we are cutting off two squares, so values <em>w<\/em>\u00a0may take on are greater than zero or less than 7. This means we will restrict the domain of this function to [latex]0&lt;w&lt;7[\/latex]. Using technology to sketch the graph of [latex]V\\left(w\\right)[\/latex] on this reasonable domain, we get a graph like Figure 24. We can use this graph to estimate the maximum value for the volume, restricted to values for <em>w<\/em>\u00a0that are reasonable for this problem\u2014values from 0 to 7.<span id=\"fs-id1165137852816\">\r\n<\/span><\/p>\r\nFrom this graph, we turn our focus to only the portion on the reasonable domain, [latex]\\left[0,\\text{ }7\\right][\/latex]. We can estimate the maximum value to be around 340 cubic cm, which occurs when the squares are about 2.75 cm on each side. To improve this estimate, we could use advanced features of our technology, if available, or simply change our window to zoom in on our graph to produce the graph in Figure 24.\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\/03010736\/CNX_Precalc_Figure_03_04_0292.jpg\" alt=\"Graph of V(w)=(20-2w)(14-2w)w where the x-axis is labeled w and the y-axis is labeled V(w) on the domain [2.4, 3].\" width=\"487\" height=\"444\" \/> <b>Figure 24<\/b>[\/caption]\r\n<p id=\"fs-id1165133036028\">From this zoomed-in view, we can refine our estimate for the maximum volume to about 339 cubic cm, when the squares measure approximately 2.7 cm on each side.<\/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-id1165134199283\">Use technology to find the maximum and minimum values on the interval [latex]\\left[-1,4\\right][\/latex] of the function [latex]f\\left(x\\right)=-0.2{\\left(x - 2\\right)}^{3}{\\left(x+1\\right)}^{2}\\left(x - 4\\right)[\/latex].<\/p>\r\n[reveal-answer q=\"747502\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"747502\"]\r\n\r\nThe minimum occurs at approximately the point [latex]\\left(0,-6.5\\right)[\/latex], and the maximum occurs at approximately the point [latex]\\left(3.5,7\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>\r\n<h2>Key Concepts<\/h2>\r\n<ul id=\"fs-id1165137846272\">\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.<\/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 \t<li>The graph of a polynomial function changes direction at its turning points.<\/li>\r\n \t<li>A polynomial function of degree <em>n<\/em>\u00a0has at most\u00a0<em>n <\/em>\u2013\u00a01 turning points.<\/li>\r\n \t<li>To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most<em>\u00a0n <\/em>\u2013\u00a01 turning points.<\/li>\r\n \t<li>Graphing a polynomial function helps to estimate local and global extremas.<\/li>\r\n \t<li>The Intermediate Value Theorem tells us that if [latex]f\\left(a\\right) \\text{and} f\\left(b\\right)[\/latex]\u00a0have opposite signs, then there exists at least one value <em>c<\/em>\u00a0between <em>a<\/em>\u00a0and <em>b<\/em>\u00a0for which [latex]f\\left(c\\right)=0[\/latex].<\/li>\r\n<\/ul>\r\n<div>\r\n<h2>Glossary<\/h2>\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-id1165135528510\" class=\"definition\">\r\n \t<dt><strong>Intermediate Value Theorem<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135528515\">for two numbers <em>a<\/em>\u00a0and <em>b<\/em>\u00a0in the domain of <em>f<\/em>,\u00a0if [latex]a&lt;b[\/latex]\u00a0and [latex]f\\left(a\\right)\\ne f\\left(b\\right)[\/latex],\u00a0then the function <em>f<\/em>\u00a0takes on every value between [latex]f\\left(a\\right)[\/latex]\u00a0and [latex]f\\left(b\\right)[\/latex];\u00a0specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the <em>x<\/em>-axis<\/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<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li style=\"font-weight: 400;\">Recognize characteristics of graphs of polynomial functions.<\/li>\n<li style=\"font-weight: 400;\">Identify zeros of polynomials and their multiplicities.<\/li>\n<li style=\"font-weight: 400;\">Determine end behavior.<\/li>\n<li style=\"font-weight: 400;\">Understand the relationship between degree and turning points.<\/li>\n<li style=\"font-weight: 400;\">Graph polynomial functions.<\/li>\n<li style=\"font-weight: 400;\">Solve polynomial inequalities.<\/li>\n<li style=\"font-weight: 400;\">Use the Intermediate Value Theorem.<\/li>\n<li>Write the formula for a polynomial function.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165135545777\">The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in the table below<b>.<\/b><\/p>\n<table id=\"Table_03_04_01\" summary=\"Two rows and nine columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<td>2013<\/td>\n<\/tr>\n<tr>\n<td><strong>Revenues<\/strong><\/td>\n<td>52.4<\/td>\n<td>52.8<\/td>\n<td>51.2<\/td>\n<td>49.5<\/td>\n<td>48.6<\/td>\n<td>48.6<\/td>\n<td>48.7<\/td>\n<td>47.1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165134040487\">The revenue can be modeled by the polynomial function<\/p>\n<div id=\"eip-679\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]R\\left(t\\right)=-0.037{t}^{4}+1.414{t}^{3}-19.777{t}^{2}+118.696t - 205.332[\/latex]<\/div>\n<p id=\"fs-id1165137659450\">where <em>R<\/em>\u00a0represents the revenue in millions of dollars and <em>t<\/em>\u00a0represents the year, with <em>t<\/em> = 6\u00a0corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.<\/p>\n<h2>Recognize characteristics of graphs of polynomial functions<\/h2>\n<p id=\"fs-id1165134352567\">Polynomial functions of degree 2 or more have graphs that do not have sharp corners; recall that these types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. Figure 1 shows\u00a0a graph that represents a <strong>polynomial function<\/strong> and a graph that represents a function that is not a polynomial.<span id=\"fs-id1165135185916\"><br \/>\n<\/span><\/p>\n<div style=\"width: 910px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" 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_0012.jpg\" alt=\"Graph of f(x)=x^3-0.01x.\" width=\"900\" height=\"409\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1<\/b><\/p>\n<\/div>\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 1: Recognizing Polynomial Functions<\/h3>\n<p>Which of the graphs in Figure 2\u00a0represents a 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 2<\/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-id1165134129608\">The graphs of <em>f<\/em>\u00a0and <em>h<\/em>\u00a0are graphs of polynomial functions. They are smooth and <strong>continuous<\/strong>.<\/p>\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<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134164967\" class=\"note precalculus qa textbox\">\n<h3>Q &amp; A<\/h3>\n<p id=\"fs-id1165135496631\"><strong>Do all polynomial functions have as their domain all real numbers?<\/strong><\/p>\n<p id=\"fs-id1165134342693\"><em>Yes. Any real number is a valid input for a polynomial function.<\/em><\/p>\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 2: 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 3<\/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 3: 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 4<\/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 4: 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 5.<\/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 5<\/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 5: 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 6, 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 6<\/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 2<\/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>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 7\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 7.<\/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 8\u00a0for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.<\/p>\n<div id=\"attachment_16092\" style=\"width: 884px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/courses.lumenlearning.com\/precalculus\/wp-content\/uploads\/sites\/3675\/2018\/08\/404d5117e8c2b2cc187c001d0fcf267e8d3c7bbf.jpeg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-16092\" class=\"wp-image-16092 size-full\" src=\"https:\/\/courses.lumenlearning.com\/precalculus\/wp-content\/uploads\/sites\/3675\/2018\/08\/404d5117e8c2b2cc187c001d0fcf267e8d3c7bbf.jpeg\" alt=\"Three graphs, left to right, with zeros of multiplicity 1, 2, and 3.\" width=\"874\" height=\"324\" srcset=\"https:\/\/courses.lumenlearning.com\/precalculus\/wp-content\/uploads\/sites\/3675\/2018\/08\/404d5117e8c2b2cc187c001d0fcf267e8d3c7bbf.jpeg 874w, https:\/\/courses.lumenlearning.com\/precalculus\/wp-content\/uploads\/sites\/3675\/2018\/08\/404d5117e8c2b2cc187c001d0fcf267e8d3c7bbf-300x111.jpeg 300w, https:\/\/courses.lumenlearning.com\/precalculus\/wp-content\/uploads\/sites\/3675\/2018\/08\/404d5117e8c2b2cc187c001d0fcf267e8d3c7bbf-768x285.jpeg 768w, https:\/\/courses.lumenlearning.com\/precalculus\/wp-content\/uploads\/sites\/3675\/2018\/08\/404d5117e8c2b2cc187c001d0fcf267e8d3c7bbf-65x24.jpeg 65w, https:\/\/courses.lumenlearning.com\/precalculus\/wp-content\/uploads\/sites\/3675\/2018\/08\/404d5117e8c2b2cc187c001d0fcf267e8d3c7bbf-225x83.jpeg 225w, https:\/\/courses.lumenlearning.com\/precalculus\/wp-content\/uploads\/sites\/3675\/2018\/08\/404d5117e8c2b2cc187c001d0fcf267e8d3c7bbf-350x130.jpeg 350w\" sizes=\"auto, (max-width: 874px) 100vw, 874px\" \/><\/a><\/p>\n<p id=\"caption-attachment-16092\" class=\"wp-caption-text\">Figure 8.<\/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 6: 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 9<\/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 9 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 10<\/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 3, a zero of \u20131 with multiplicity 2, and a zero of 3 with even multiplicity of 4.<\/p>\n<\/div>\n<\/div>\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>Understand the relationship between degree and turning points<\/h2>\n<p id=\"fs-id1165135416524\">In addition to the end behavior, recall that we can analyze a polynomial function\u2019s local behavior. It may have a turning point where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). Look at the graph of the polynomial function [latex]f\\left(x\\right)={x}^{4}-{x}^{3}-4{x}^{2}+4x[\/latex] in Figure 11. 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 11<\/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 7: 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<h2>\u00a0Graph polynomial functions<\/h2>\n<p id=\"fs-id1165137843095\">We can use what we have learned about multiplicities, end behavior, and turning points to sketch graphs of polynomial functions. Let us put this all together and look at the steps required to graph polynomial functions.<\/p>\n<div id=\"fs-id1165137843101\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135449677\">How To: Given a polynomial function, sketch the graph.<\/h3>\n<ol id=\"fs-id1165135449683\">\n<li>Find the intercepts.<\/li>\n<li>Check for symmetry. If the function is an even function, its graph is symmetrical about the <em>y<\/em>-axis, that is,\u00a0<em>f<\/em>(\u2013<em>x<\/em>) = <em>f<\/em>(<em>x<\/em>).<br \/>\nIf a function is an odd function, its graph is symmetrical about the origin, that is,\u00a0<em>f<\/em>(\u2013<em>x<\/em>) = <em>\u2013<\/em><em>f<\/em>(<em>x<\/em>).<\/li>\n<li>Use the multiplicities of the zeros to determine the behavior of the polynomial at the <em>x<\/em>-intercepts.<\/li>\n<li>Determine the end behavior by examining the leading term.<\/li>\n<li>Use the end behavior and the behavior at the intercepts to sketch a graph.<\/li>\n<li>Ensure that the number of turning points does not exceed one less than the degree of the polynomial.<\/li>\n<li>Optionally, use technology to check the graph.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_04_08\" class=\"example\">\n<div id=\"fs-id1165135575951\" class=\"exercise\">\n<div id=\"fs-id1165135575953\" class=\"problem textbox shaded\">\n<h3>Example 8: Sketching the Graph of a Polynomial Function<\/h3>\n<p id=\"fs-id1165135575958\">Sketch a graph of [latex]f\\left(x\\right)=-2{\\left(x+3\\right)}^{2}\\left(x - 5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q892446\">Show Solution<\/span><\/p>\n<div id=\"q892446\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135237929\">This graph has two <em>x-<\/em>intercepts. At <em>x\u00a0<\/em>= \u20133, the factor is squared, indicating a multiplicity of 2. The graph will bounce at this <em>x<\/em>-intercept. At <em>x\u00a0<\/em>= 5, the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept.<\/p>\n<p id=\"fs-id1165135171021\">The <em>y<\/em>-intercept is found by evaluating <em>f<\/em>(0).<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} f\\left(0\\right)&=-2{\\left(0+3\\right)}^{2}\\left(0 - 5\\right) \\\\ &=-2\\cdot 9\\cdot \\left(-5\\right) \\\\ &=90 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165134374772\">The <em>y<\/em>-intercept is (0, 90).<\/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_0172.jpg\" alt=\"Showing the distribution for the leading term.\" width=\"487\" height=\"362\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 13<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165134381522\">Additionally, we can see the leading term, if this polynomial were multiplied out, would be [latex]-2{x}^{3}[\/latex],<br \/>\nso the end behavior is that of a vertically reflected cubic, with the outputs decreasing as the inputs approach infinity, and the outputs increasing as the inputs approach negative infinity.<span id=\"fs-id1165135646080\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165134374738\">To sketch this, we consider that:<\/p>\n<ul id=\"fs-id1165134374741\">\n<li>As [latex]x\\to -\\infty[\/latex] the function [latex]f\\left(x\\right)\\to \\infty[\/latex], so we know the graph starts in the second quadrant and is decreasing toward the <em>x<\/em>-axis.<\/li>\n<li>Since [latex]f\\left(-x\\right)=-2{\\left(-x+3\\right)}^{2}\\left(-x - 5\\right)[\/latex]<br \/>\nis not equal to <em>f<\/em>(<em>x<\/em>), the graph does not display symmetry.<\/li>\n<li>At (-3,0), the graph bounces off of the <em>x<\/em>-axis, so the function must start increasing.\n<p id=\"fs-id1165135536183\" style=\"text-align: left;\">At (0, 90), the graph crosses the <em>y<\/em>-axis at the <em>y<\/em>-intercept.<\/p>\n<\/li>\n<\/ul>\n<figure id=\"Figure_03_04_018\" class=\"small\">\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_0182.jpg\" alt=\"Graph of the end behavior and intercepts, (-3, 0) and (0, 90), for the function f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"362\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 14<\/b><\/p>\n<\/div>\n<\/figure>\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\/03010734\/CNX_Precalc_Figure_03_04_0192.jpg\" alt=\"Graph of the end behavior and intercepts, (-3, 0), (0, 90) and (5, 0), for the function f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"362\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 15<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165135241000\">Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the graph passes through the next intercept at (5, 0).\u00a0<span id=\"fs-id1165135241013\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165135613608\">As [latex]x\\to \\infty[\/latex] the function [latex]f\\left(x\\right)\\to \\mathrm{-\\infty }[\/latex], so we know the graph continues to decrease, and we can stop drawing the graph in the fourth quadrant.<\/p>\n<p id=\"fs-id1165135574296\">Using technology, we can create the graph for the polynomial function, shown in Figure 16, and verify that the resulting graph looks like our sketch in Figure 15.<\/p>\n<figure id=\"Figure_03_04_020\" class=\"small\"><figcaption>The complete graph of the polynomial function [latex]f\\left(x\\right)=-2{\\left(x+3\\right)}^{2}\\left(x - 5\\right)[\/latex]<\/figcaption><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\/03010734\/CNX_Precalc_Figure_03_04_0202.jpg\" alt=\"Graph of f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"366\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 16<\/b><\/p>\n<\/div>\n<\/figure>\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-id1165133065140\">Sketch a graph of [latex]f\\left(x\\right)=\\frac{1}{4}x{\\left(x - 1\\right)}^{4}{\\left(x+3\\right)}^{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q408253\">Show Solution<\/span><\/p>\n<div id=\"q408253\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010734\/CNX_Precalc_Figure_03_04_0212.jpg\" alt=\"Graph of f(x)=(1\/4)x(x-1)^4(x+3)^3.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>\u00a0Solving Polynomial Inequalities<\/h2>\n<p>One application of our ability to find intercepts and sketch a graph of polynomials is the ability to solve polynomial inequalities. It is a very common question to ask when a function will be positive and negative. We can solve polynomial inequalities by either utilizing the graph, or by using test values.<\/p>\n<div id=\"fs-id1165137433651\" class=\"solution textbox shaded\">\n<h3>Example 9: Solving Polynomial Inequalities in Factored From<\/h3>\n<p>Solve [latex]\\left(x+3\\right){\\left(x+1\\right)}^{2}\\left(x-4\\right)> 0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q494744\">Show Solution<\/span><\/p>\n<div id=\"q494744\" class=\"hidden-answer\" style=\"display: none\">\n<p>As with all inequalities, we start by solving the equality [latex]\\left(x+3\\right){\\left(x+1\\right)}^{2}\\left(x-4\\right)= 0[\/latex], which has solutions at x = -3, -1, and 4. We know the function can only change from positive to negative at these values, so these divide the inputs into 4 intervals.<br \/>\nWe could choose a test value in each interval and evaluate the function [latex]f\\left(x\\right) = \\left(x+3\\right){\\left(x+1\\right)}^{2}\\left(x-4\\right)[\/latex] at each test value to determine if the function is positive or negative in that interval<\/p>\n<table>\n<tbody>\n<tr>\n<td>Interval<\/td>\n<td>Test x in interval<\/td>\n<td>f(test value)<\/td>\n<td>&gt; 0 or &lt; 0<\/td>\n<\/tr>\n<tr>\n<td>x &lt; -3<\/td>\n<td>-4<\/td>\n<td>72<\/td>\n<td>&gt; 0<\/td>\n<\/tr>\n<tr>\n<td>-3 &lt; x &lt; -1<\/td>\n<td>-2<\/td>\n<td>-6<\/td>\n<td>&lt; 0<\/td>\n<\/tr>\n<tr>\n<td>-1 &lt;\u00a0 x &lt; 4<\/td>\n<td>0<\/td>\n<td>-12<\/td>\n<td>&lt; 0<\/td>\n<\/tr>\n<tr>\n<td>x &gt; 4<\/td>\n<td>5<\/td>\n<td>288<\/td>\n<td>\u00a0&gt; 0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>On a number line this would look like:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-13403 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/139\/2016\/04\/28183410\/1.png\" alt=\"Number line with values from -6 to 6 double headed arrows from -6 to -3 read positive, from -3 to -1 read negative, from -1 to positive 4 read negative and from 4 to 6 read positive.\" width=\"630\" height=\"110\" \/><br \/>\nFrom our test values, we can determine this function is positive when <em>x<\/em> &lt; -3 or <em>x<\/em> &gt; 4, or in interval notation, [latex]\\left(-\\infty, -3\\right)\\cup\\left(4,\\infty\\right)[\/latex]. We could have also determined on which intervals the function was positive by sketching a graph of the function. We illustrate that technique in the next example.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137433651\" class=\"solution textbox shaded\">\n<h3>Example 10: Solving Polynomial Inequalities in Factored From<\/h3>\n<p>Find the domain of the function [latex]v\\left(t\\right)=\\sqrt{6-5t-{t}^{2}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q359527\">Show Solution<\/span><\/p>\n<div id=\"q359527\" class=\"hidden-answer\" style=\"display: none\">\n<p>A square root is only defined when the quantity we are taking the square root of, the quantity inside the square root, is zero or greater. Thus, the domain of this function will be when [latex]6 - 5t - {t}^{2}\\ge 0[\/latex]. Again we start by solving the equality [latex]6 - 5t - {t}^{2}= 0[\/latex]. While we could use the quadratic formula, this equation factors nicely to [latex]\\left(6 + t\\right)\\left(1-t\\right)=0[\/latex], giving horizontal intercepts<br \/>\nt = 1 and t = -6.<br \/>\nSketching a graph of this quadratic will allow us to determine when it is positive.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-13404\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/139\/2016\/04\/28183442\/Screen-Shot2.png\" alt=\"Graph of upside down parabola on cartesian coordinate axes passing through (-6,0) and (1,0)\" width=\"278\" height=\"204\" \/><br \/>\nFrom the graph we can see this function is positive for inputs between the intercepts. So [latex]6 - 5t - {t}^{2}\\ge 0[\/latex] is positive for [latex]-6 \\le t\\le 1[\/latex], and this will be the domain of the v(t) function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137433651\" class=\"solution textbox shaded\">\n<h3>Example 11: Solving a Polynomial Inequality Not in Factored Form<\/h3>\n<p>Solve the inequality [latex]{x}^{4} - 2{x}^{3} - 3{x}^{2} \\gt 0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q273617\">Show Solution<\/span><\/p>\n<div id=\"q273617\" class=\"hidden-answer\" style=\"display: none\">\n<p>In our other examples, we were given polynomials that were already in factored form, here we have an additional step to finding the intervals on which solutions to the given inequality lie. Again, we will start by solving the equality [latex]{x}^{4} - 2{x}^{3} - 3{x}^{2} = 0[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left;\">Notice that there is a common factor of [latex]{x}^{2}[\/latex] in each term of this polynomial. We can use factoring to simplify in the following way:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}{x}^{4} - 2{x}^{3} - 3{x}^{2} &= 0&\\\\{x}^{2}\\left({x}^{2} - 2{x} - 3\\right) &= 0\\\\ {x}^{2}\\left(x - 3\\right)\\left(x + 1 \\right)&= 0\\end{align}[\/latex]<\/p>\n<p>Now we can set each factor equal to zero to find the solution to the equality.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc} {x}^{2} = 0 & \\left(x - 3\\right) = 0 &\\left(x+1\\right) = 0\\\\ {x} = 0 & x = 3 & x = -1\\\\ \\end{array}[\/latex].<\/p>\n<p>Note that x = 0 has multiplicity of two, but since our inequality is strictly greater than, we don&#8217;t need to include it in our solutions.<br \/>\nWe can choose a test value in each interval and evaluate the function<\/p>\n<p style=\"text-align: center;\">[latex]{x}^{4} - 2{x}^{3} - 3{x}^{2} = 0[\/latex]<\/p>\n<p style=\"text-align: left;\">at each test value to determine if the function is positive or negative in that interval<\/p>\n<table>\n<tbody>\n<tr>\n<td>Interval<\/td>\n<td>Test x in interval<\/td>\n<td>&gt; 0,\u00a0 &lt; 0<\/td>\n<\/tr>\n<tr>\n<td>x &lt; -1<\/td>\n<td>-2<\/td>\n<td>x &gt; 0<\/td>\n<\/tr>\n<tr>\n<td>-1 &lt; x &lt; 0<\/td>\n<td>-1\/2<\/td>\n<td>\u00a0x &lt;\u00a0 0<\/td>\n<\/tr>\n<tr>\n<td>0 &lt; x &lt; 3<\/td>\n<td>1<\/td>\n<td>x &lt; 0<\/td>\n<\/tr>\n<tr>\n<td>x &gt; 3<\/td>\n<td>5<\/td>\n<td>x &gt; 0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We want to have the set of x values that will give us the intervals where the polynomial is greater than zero. Our answer will be [latex]\\left(-\\infty, -1\\right]\\cup\\left[3,\\infty\\right)[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>The graph of the function gives us additional confirmation of our solution.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-13406 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/139\/2016\/04\/28183505\/Screen-Shot3.png\" alt=\"Line dips down, dips slightly up, dips very far down, then sharply goes up\" width=\"302\" height=\"445\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it 5<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm34324\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=34324&theme=oea&iframe_resize_id=ohm34324&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Further Examples<\/h2>\n<h3>Solving a polynomial inequality not in factored form &#8211; use factoring by grouping.<\/h3>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Solve a Polynomial Inequality - Factor By Grouping (Degree 3)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/jmeLkQCFLHs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Solving a polynomial inequality not in factored form &#8211; use greatest common factor.<\/h3>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Solve a Polynomial Inequality - Factor Using GCF (Degree 3)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/zyiad-T6-TI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Solving a polynomial inequality not in factored form &#8211; factor a trinomial<\/h3>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex: Solve a Polynomial Inequality - Factor a Trinomial (Degree 4)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/LC1bwRHcdh4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Use the Intermediate Value Theorem<\/h2>\n<p id=\"fs-id1165135205093\">In some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the <em>x<\/em>-axis, we can confirm that there is a zero between them. Consider a polynomial function <em>f<\/em>\u00a0whose graph is smooth and continuous. The <strong>Intermediate Value Theorem<\/strong> states that for two numbers <em>a<\/em>\u00a0and <em>b<\/em>\u00a0in the domain of <em>f<\/em>,\u00a0if <em>a\u00a0<\/em>&lt; <em>b<\/em>\u00a0and [latex]f\\left(a\\right)\\ne f\\left(b\\right)[\/latex], then the function <em>f<\/em>\u00a0takes on every value between [latex]f\\left(a\\right)[\/latex] and [latex]f\\left(b\\right)[\/latex].<\/p>\n<p>We can apply this theorem to a special case that is useful in graphing polynomial functions. If a point on the graph of a continuous function <em>f<\/em>\u00a0at [latex]x=a[\/latex] lies above the <em>x<\/em>-axis and another point at [latex]x=b[\/latex] lies below the <em>x<\/em>-axis, there must exist a third point between [latex]x=a[\/latex] and [latex]x=b[\/latex] where the graph crosses the <em>x<\/em>-axis. Call this point [latex]\\left(c,\\text{ }f\\left(c\\right)\\right)[\/latex]. This means that we are assured there is a solution <em>c<\/em>\u00a0where [latex]f\\left(c\\right)=0[\/latex].<\/p>\n<p>In other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the <em>x<\/em>-axis. Figure 17\u00a0shows that there is a zero between <em>a<\/em>\u00a0and <em>b<\/em>.<\/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\/03010734\/CNX_Precalc_Figure_03_04_0222.jpg\" alt=\"Graph of an odd-degree polynomial function that shows a point f(a) that\u2019s negative, f(b) that\u2019s positive, and f(c) that\u2019s 0.\" width=\"487\" height=\"368\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 17.<\/b> Using the Intermediate Value Theorem to show there exists a zero.<\/p>\n<\/div>\n<div id=\"fs-id1165135347510\" class=\"note textbox shaded\">\n<h3 class=\"title\">A General Note: Intermediate Value Theorem<\/h3>\n<p id=\"fs-id1165135580347\">Let <em>f<\/em>\u00a0be a polynomial function. The <strong>Intermediate Value Theorem<\/strong> states that if [latex]f\\left(a\\right)[\/latex]\u00a0and [latex]f\\left(b\\right)[\/latex]\u00a0have opposite signs, then there exists at least one value <em>c<\/em>\u00a0between <em>a<\/em>\u00a0and <em>b<\/em>\u00a0for which [latex]f\\left(c\\right)=0[\/latex].<\/p>\n<\/div>\n<div id=\"Example_03_04_09\" class=\"example\">\n<div id=\"fs-id1165133358799\" class=\"exercise\">\n<div id=\"fs-id1165133358801\" class=\"problem textbox shaded\">\n<h3>Example 12: Using the Intermediate Value Theorem<\/h3>\n<p id=\"fs-id1165133358807\">Show that the function [latex]f\\left(x\\right)={x}^{3}-5{x}^{2}+3x+6[\/latex]\u00a0has at least two real zeros between [latex]x=1[\/latex]\u00a0and [latex]x=4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q98939\">Show Solution<\/span><\/p>\n<div id=\"q98939\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135537349\">As a start, evaluate [latex]f\\left(x\\right)[\/latex]\u00a0at the integer values [latex]x=1,2,3,\\text{ and }4[\/latex].<\/p>\n<table id=\"Table_03_04_03\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td><em><strong>f\u00a0<\/strong><\/em><strong>(<em>x<\/em>)<\/strong><\/td>\n<td>5<\/td>\n<td>0<\/td>\n<td>\u20133<\/td>\n<td>2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135536378\">We see that one zero occurs at [latex]x=2[\/latex]. Also, since [latex]f\\left(3\\right)[\/latex] is negative and [latex]f\\left(4\\right)[\/latex] is positive, by the Intermediate Value Theorem, there must be at least one real zero between 3 and 4.<\/p>\n<p id=\"fs-id1165135575934\">We have shown that there are at least two real zeros between [latex]x=1[\/latex]\u00a0and [latex]x=4[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We can also see in Figure 18\u00a0that there are two real zeros between [latex]x=1[\/latex]\u00a0and [latex]x=4[\/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\/03010735\/CNX_Precalc_Figure_03_04_0232.jpg\" alt=\"Graph of f(x)=x^3-5x^2+3x+6 and shows, by the Intermediate Value Theorem, that there exists two zeros since f(1)=5 and f(4)=2 are positive and f(3) = -3 is negative.\" width=\"487\" height=\"591\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 18<\/b><\/p>\n<\/div>\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-id1165135551168\">Show that the function [latex]f\\left(x\\right)=7{x}^{5}-9{x}^{4}-{x}^{2}[\/latex] has at least one real zero between [latex]x=1[\/latex] and [latex]x=2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q886741\">Show Solution<\/span><\/p>\n<div id=\"q886741\" class=\"hidden-answer\" style=\"display: none\">\n<p>Because <em>f<\/em>\u00a0is a polynomial function and since [latex]f\\left(1\\right)[\/latex] is negative and [latex]f\\left(2\\right)[\/latex] is positive, there is at least one real zero between [latex]x=1[\/latex] and [latex]x=2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<section id=\"fs-id1165135369116\">\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 19.<\/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 19<\/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 20, 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 20<\/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<\/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 21.<\/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 21<\/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<div id=\"Example_03_04_11\" class=\"example\">\n<div id=\"fs-id1165135470044\" class=\"exercise\">\n<div id=\"fs-id1165135470046\" class=\"problem textbox shaded\">\n<h3>Example 14: Using Local Extrema to Solve Applications<\/h3>\n<p id=\"fs-id1165135470052\">An open-top box is to be constructed by cutting out squares from each corner of a 14 cm by 20 cm sheet of plastic then folding up the sides. Find the size of squares that should be cut out to maximize the volume enclosed by the box.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q850359\">Show Solution<\/span><\/p>\n<div id=\"q850359\" class=\"hidden-answer\" style=\"display: none\">\n<p>We will start this problem by drawing a picture like Figure 22, labeling the width of the cut-out squares with a variable, <em>w<\/em>.<\/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\/03010736\/CNX_Precalc_Figure_03_04_0272.jpg\" alt=\"Diagram of a rectangle with four squares at the corners.\" width=\"487\" height=\"298\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 22<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165134150016\">Notice that after a square is cut out from each end, it leaves a [latex]\\left(14 - 2w\\right)[\/latex] cm by [latex]\\left(20 - 2w\\right)[\/latex] cm rectangle for the base of the box, and the box will be <em>w<\/em>\u00a0cm tall. This gives the volume<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}V\\left(w\\right)&=\\left(20 - 2w\\right)\\left(14 - 2w\\right)w \\\\ &=280w - 68{w}^{2}+4{w}^{3} \\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\/03010736\/CNX_Precalc_Figure_03_04_0282.jpg\" alt=\"Graph of V(w)=(20-2w)(14-2w)w where the x-axis is labeled w and the y-axis is labeled V(w).\" width=\"487\" height=\"406\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 23<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165135628578\">Notice, since the factors are <em>w<\/em>, [latex]20 - 2w[\/latex] and [latex]14 - 2w[\/latex], the three zeros are 10, 7, and 0, respectively. Because a height of 0 cm is not reasonable, we consider the only the zeros 10 and 7. The shortest side is 14 and we are cutting off two squares, so values <em>w<\/em>\u00a0may take on are greater than zero or less than 7. This means we will restrict the domain of this function to [latex]0<w<7[\/latex]. Using technology to sketch the graph of [latex]V\\left(w\\right)[\/latex] on this reasonable domain, we get a graph like Figure 24. We can use this graph to estimate the maximum value for the volume, restricted to values for <em>w<\/em>\u00a0that are reasonable for this problem\u2014values from 0 to 7.<span id=\"fs-id1165137852816\"><br \/>\n<\/span><\/p>\n<p>From this graph, we turn our focus to only the portion on the reasonable domain, [latex]\\left[0,\\text{ }7\\right][\/latex]. We can estimate the maximum value to be around 340 cubic cm, which occurs when the squares are about 2.75 cm on each side. To improve this estimate, we could use advanced features of our technology, if available, or simply change our window to zoom in on our graph to produce the graph in Figure 24.<\/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\/03010736\/CNX_Precalc_Figure_03_04_0292.jpg\" alt=\"Graph of V(w)=(20-2w)(14-2w)w where the x-axis is labeled w and the y-axis is labeled V(w) on the domain &#091;2.4, 3&#093;.\" width=\"487\" height=\"444\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 24<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165133036028\">From this zoomed-in view, we can refine our estimate for the maximum volume to about 339 cubic cm, when the squares measure approximately 2.7 cm on each side.<\/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-id1165134199283\">Use technology to find the maximum and minimum values on the interval [latex]\\left[-1,4\\right][\/latex] of the function [latex]f\\left(x\\right)=-0.2{\\left(x - 2\\right)}^{3}{\\left(x+1\\right)}^{2}\\left(x - 4\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q747502\">Show Solution<\/span><\/p>\n<div id=\"q747502\" class=\"hidden-answer\" style=\"display: none\">\n<p>The minimum occurs at approximately the point [latex]\\left(0,-6.5\\right)[\/latex], and the maximum occurs at approximately the point [latex]\\left(3.5,7\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165137846272\">\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.<\/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<li>The graph of a polynomial function changes direction at its turning points.<\/li>\n<li>A polynomial function of degree <em>n<\/em>\u00a0has at most\u00a0<em>n <\/em>\u2013\u00a01 turning points.<\/li>\n<li>To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most<em>\u00a0n <\/em>\u2013\u00a01 turning points.<\/li>\n<li>Graphing a polynomial function helps to estimate local and global extremas.<\/li>\n<li>The Intermediate Value Theorem tells us that if [latex]f\\left(a\\right) \\text{and} f\\left(b\\right)[\/latex]\u00a0have opposite signs, then there exists at least one value <em>c<\/em>\u00a0between <em>a<\/em>\u00a0and <em>b<\/em>\u00a0for which [latex]f\\left(c\\right)=0[\/latex].<\/li>\n<\/ul>\n<div>\n<h2>Glossary<\/h2>\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-id1165135528510\" class=\"definition\">\n<dt><strong>Intermediate Value Theorem<\/strong><\/dt>\n<dd id=\"fs-id1165135528515\">for two numbers <em>a<\/em>\u00a0and <em>b<\/em>\u00a0in the domain of <em>f<\/em>,\u00a0if [latex]a<b[\/latex]\u00a0and [latex]f\\left(a\\right)\\ne f\\left(b\\right)[\/latex],\u00a0then the function <em>f<\/em>\u00a0takes on every value between [latex]f\\left(a\\right)[\/latex]\u00a0and [latex]f\\left(b\\right)[\/latex];\u00a0specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the <em>x<\/em>-axis<\/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<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-13852\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":97803,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-13852","chapter","type-chapter","status-publish","hentry"],"part":10733,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/13852","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/97803"}],"version-history":[{"count":11,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/13852\/revisions"}],"predecessor-version":[{"id":16093,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/13852\/revisions\/16093"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/10733"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/13852\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=13852"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=13852"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=13852"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=13852"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}