{"id":238,"date":"2024-10-18T21:13:19","date_gmt":"2024-10-18T21:13:19","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/polynomial-functions-fresh-take\/"},"modified":"2024-10-18T21:25:14","modified_gmt":"2024-10-18T21:25:14","slug":"polynomial-functions-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/polynomial-functions-fresh-take\/","title":{"raw":"Polynomial Functions: Fresh Take","rendered":"Polynomial Functions: Fresh Take"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Recognize polynomial functions, noting their degree and leading coefficient<\/li>\n\t<li>Apply various methods to divide polynomials and locate the zeros of polynomial equations<\/li>\n\t<li>Generate graphs and formulate equations for polynomial functions<\/li>\n\t<li>Explain the end behavior of polynomial functions<\/li>\n<\/ul>\n<\/section>\n<h2>Identifying Polynomial Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>Polynomial functions are like the DNA of algebra\u2014they combine simplicity and complexity to form an incredibly diverse array of functions.<\/p>\n<p>At their core, polynomials are sums of terms made up of coefficients and variables raised to whole number powers. The degree of a polynomial, given by the highest power of the variable, tells us a lot about the function's behavior and the shape of its graph.<\/p>\n<p>Let [latex]n[\/latex]&nbsp; be a non-negative integer. A <strong>polynomial function<\/strong> is a function that can be written in the form<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/p>\n<p>This is called the <strong>general form of a polynomial function<\/strong>.<\/p>\n<p>Each [latex]{a}_{i}[\/latex]&nbsp;is a coefficient and can be any real number.<\/p>\n<p>Each product [latex]{a}_{i}{x}^{i}[\/latex]&nbsp;is a <strong>term of a polynomial function<\/strong>.<br>\n<br>\n<strong>Quick Tips<\/strong><\/p>\n<ul>\n\t<li><strong>Linear, Quadratic, and Beyond:<\/strong> A first-degree polynomial is linear, a second-degree is quadratic, and higher degrees have their own characteristics and complexities.<\/li>\n\t<li><strong>Coefficient Clues:<\/strong> The coefficients in a polynomial can tell us about the steepness and direction of the graph. A positive leading coefficient means the graph opens upward, and a negative one indicates it opens downward.<\/li>\n<\/ul>\n<\/div>\n<h2>Defining the Degree and Leading Coefficient of a Polynomial Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n\nThe degree of a polynomial is the highest power of the variable present. The leading coefficient is the coefficient of the term with the highest power. These two characteristics can tell us a lot about the function's behavior, especially its growth and end behavior.\n\n<ul>\n\t<li>\n<p><strong>Degree Tells the Tale:<\/strong> The degree of a polynomial function hints at the number of roots and the possible number of turns on its graph. For instance, a second-degree polynomial, or a quadratic, will have at most two roots and one turn.<\/p>\n<\/li>\n\t<li>\n<p><strong>Leading Coefficient Impact:<\/strong> The leading coefficient isn't just a number\u2014it's the boss. It influences the end behavior of the polynomial's graph. If it's positive, the graph eventually rises; if negative, the graph falls.<\/p>\n<\/li>\n<\/ul>\n<strong>Quick Tips: Identifying Degree and Leading Coefficient<\/strong>\n<ul>\n\t<li>Arrange terms in descending order of power to easily identify the leading term and coefficient.<\/li>\n\t<li>The degree gives a sense of the 'shape' of the graph of the polynomial function.<\/li>\n\t<li>For instance, [latex]f(x)=3+2x^2\u22124x^3[\/latex] has a degree of [latex]3[\/latex] and a leading coefficient of [latex]-4[\/latex], indicating a cubic function with a negative leading term.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Identify the degree, leading term, and leading coefficient of the polynomial [latex]f\\left(x\\right)=4{x}^{2}-{x}^{6}+2x - 6[\/latex].<\/p>\n<p>[reveal-answer q=\"435637\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"435637\"]<\/p>\n<p>The degree is [latex]6[\/latex]. The leading term is [latex]-{x}^{6}[\/latex]. The leading coefficient is [latex]\u20131[\/latex].[\/hidden-answer]<\/p>\n<\/section>\n<p>Watch the following video for more examples of how to determine the degree, leading term, and leading coefficient of a polynomial.<\/p>\n<section class=\"textbox watchIt\">\n<p>https:\/\/youtu.be\/F_G_w82s0QA<\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Degree%2C+Leading+Term%2C+and+Leading+Coefficient+of+a+Polynomial+Function.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDegree, Leading Term, and Leading Coefficient of a Polynomial Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Polynomial Long Division<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>Polynomial long division might seem daunting at first, but it's quite similar to the long division we learned in elementary school. Just like we divide numbers, we can divide polynomials, breaking them down into simpler parts. The process involves dividing the highest degree terms and then subtracting to find the remainder, which we then bring down the next term to continue the process.<\/p>\n<strong>Quick Tips: Steps to Polynomial Long Division<\/strong>\n<ol>\n\t<li><strong>Set Up<\/strong>: Write the polynomial (dividend) and the binomial (divisor) in long division format.<\/li>\n\t<li><strong>Divide the Leading Terms<\/strong>: Take the leading term of the dividend and divide it by the leading term of the divisor to find the first term of the quotient.<\/li>\n\t<li><strong>Multiply and Subtract<\/strong>: Multiply the entire divisor by the term just found and subtract it from the dividend.<\/li>\n\t<li><strong>Repeat<\/strong>: Bring down the next term of the dividend and repeat the process until you've worked through each term.<\/li>\n\t<li><strong>Remainder<\/strong>: If there's a remainder, express it as a fraction with the divisor as the denominator.<\/li>\n<\/ol>\n<strong>The Division Algorithm: A Formal Approach<\/strong>\n<p>The Division Algorithm is a fancy way of saying that any polynomial divided by another can be expressed as a multiplication (the quotient) plus something left over (the remainder). It's a structured method to ensure that every polynomial division can be broken down neatly.<\/p>\n<\/div>\n<section class=\"textbox example\">\n<p>Divide [latex]5{x}^{2}+3x - 2[\/latex] by [latex]x+1[\/latex].<\/p>\n<p>[reveal-answer q=\"996959\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"996959\"]<\/p>\n<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204327\/CNX_Precalc_revised_eq_22.png\" alt=\"Set up the division problem. 5x squared divided by x is 5x. Multiply x plus 1 by 5x. Subtract. Bring down the next term. Negative 2x divded by x is negative 2. Multiply x + 1 by negative 2. Subtract.\" width=\"426\" height=\"288\"><\/center>\n<p>&nbsp;<\/p>\n<p>The quotient is [latex]5x - 2[\/latex].&nbsp;The remainder is [latex]0[\/latex]. We write the result as<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{5{x}^{2}+3x - 2}{x+1}=5x - 2[\/latex]<\/p>\n<p>or<\/p>\n<p style=\"text-align: center;\">[latex]5{x}^{2}+3x - 2=\\left(x+1\\right)\\left(5x - 2\\right)[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>This division problem had a remainder of [latex]0[\/latex]. This tells us that the dividend is divided evenly by the divisor and that the divisor is a factor of the dividend.<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Divide [latex]16{x}^{3}-12{x}^{2}+20x - 3[\/latex]&nbsp;by [latex]4x+5[\/latex].<\/p>\n<p>[reveal-answer q=\"198989\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"198989\"]<\/p>\n<p>[latex]4{x}^{2}-8x+15-\\frac{78}{4x+5}[\/latex][\/hidden-answer]<\/p>\n<\/section>\n<p>Watch the following video see an example of dividing polynomials with long division.<\/p>\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328532&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=U2f5VzKGNPk&amp;video_target=tpm-plugin-yrwb7tah-U2f5VzKGNPk\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Dividing+polynomials+using+long+division.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDividing polynomials using long division\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Using Synthetic Division to Divide Polynomials<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>Synthetic division is a quicker way to divide polynomials when the divisor is a binomial with a leading coefficient of [latex]1[\/latex]. It uses only the coefficients of the polynomials, simplifying the process significantly.<\/p>\n<strong>How to Use Synthetic Division<\/strong>\n<ol>\n\t<li><strong>Write the Opposite of k<\/strong>: If dividing by [latex]x-k[\/latex], write down [latex]k[\/latex].<\/li>\n\t<li><strong>List the Coefficients<\/strong>: Write the coefficients of the polynomial you're dividing.<\/li>\n\t<li><strong>Bring Down the Leading Coefficient<\/strong>: This starts off your synthetic division.<\/li>\n\t<li><strong>Multiply and Add<\/strong>: Multiply the leading coefficient by [latex]k[\/latex], add it to the next coefficient, and continue this process.<\/li>\n\t<li><strong>Find the Quotient and Remainder<\/strong>: The last number is the remainder, and the others form the coefficients of the quotient polynomial.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Use synthetic division to divide [latex]5x^2 - 3x - 36[\/latex] by [latex]x - 3[\/latex]. [reveal-answer q=\"981343\"]Show Solution[\/reveal-answer] [hidden-answer a=\"981343\"]\n\n<p>Begin by setting up the synthetic division. Write [latex]k[\/latex] and the coefficients.<\/p>\n<center><img class=\"aligncenter wp-image-9522 size-full\" src=\"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-content\/uploads\/sites\/18\/2023\/10\/9f9b7c3f2afaf1c98578471f8ec6fd5c3b34749f.webp#fixme\" alt=\"A collapsed version of the previous synthetic division.\" width=\"487\" height=\"55\"><\/center>\n<p>&nbsp;<\/p>\n<p>Bring down the lead coefficient. Multiply the lead coefficient by [latex]k[\/latex].<\/p>\n<center><img class=\"aligncenter wp-image-9523 size-full\" src=\"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-content\/uploads\/sites\/18\/2023\/10\/8b638e9a3fafd6b1e6496583a605b4a0b4858d0e.webp#fixme\" alt=\"The set-up of the synthetic division for the polynomial 5x^2-3x-36 by x-3, which renders {5, -3, -36} by 3.\" width=\"487\" height=\"74\"><\/center>\n<p>&nbsp;<\/p>\n<p>Continue by adding the numbers in the second column. Multiply the resulting number by [latex]k[\/latex]. Write the result in the next column. Then add the numbers in the third column.<\/p>\n<center><img class=\"aligncenter wp-image-9524 size-full\" src=\"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-content\/uploads\/sites\/18\/2023\/10\/a77e7bc9a6d09d92cfe1e1f485fc8887a5ba4e6f.webp#fixme\" alt=\"Multiplied by the lead coefficient, 5, in the second column, and the lead coefficient is brought down to the second row. \" width=\"487\" height=\"74\"><\/center>\n<p>&nbsp;<\/p>\n<p>The result is [latex]5x + 12[\/latex]. The remainder is [latex]0[\/latex]. So [latex]x - 3[\/latex] is a factor of the original polynomial.<\/p>\n<strong>Analysis<\/strong>\n<p>Just as with long division, we can check our work by multiplying the quotient by the divisor and adding the remainder.<\/p>\n<center>[latex](x\u22123)(5x+12)+0=5x^2\u22123x\u221236[\/latex]<\/center>[\/hidden-answer]<\/section>\n<section class=\"textbox example\">Use synthetic division to divide [latex]-9x^4 + 10x^3 + 7x^2 - 6[\/latex] by [latex]x - 1[\/latex]. [reveal-answer q=\"981344\"]Show Solution[\/reveal-answer] [hidden-answer a=\"981344\"]\n\n<p>Notice there is no [latex]x[\/latex]-term. We will use a zero as the coefficient for that term.<\/p>\n<center><img class=\"aligncenter size-full wp-image-9533\" src=\"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-content\/uploads\/sites\/18\/2023\/09\/21c33f8114c651702043fec1fafe25f629656a9a.webp#fixme\" alt=\"A synthetic division table showing the division of a polynomial by the binomial (x - 1). The divisor '1' is on the far left. The row at the top lists the coefficients of the polynomial: -9, 10, 7, 0, and -6. The second row represents the products of the divisor and the results of each step: -9, 1, 8, and 8. The bottom row sums the coefficients and the products: -9, 1, 8, 8, and 2.\" width=\"205\" height=\"76\"><\/center>\n<p>&nbsp;<\/p>\n<p>The result is [latex]-9x^3 + x^2 + 8x + 8 + \\frac{2}{x-1}[\/latex].<\/p>\n\n[\/hidden-answer]<\/section>\n<p>Watch the following video see an example of dividing polynomials using synthetic division.<\/p>\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328533&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=FxHWoUOq2iQ&amp;video_target=tpm-plugin-6i7czt2l-FxHWoUOq2iQ\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Synthetic+Division+of+Polynomials.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSynthetic Division of Polynomials\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Evaluate Polynomials<\/h2>\n<h3>Evaluating a Polynomial Using the Remainder Theorem<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>The Remainder Theorem is a quick way to evaluate polynomials. If you divide a polynomial [latex] f(x) [\/latex]by [latex] x - k [\/latex], the remainder is simply the value of [latex] f(k) [\/latex]. It's like a shortcut in a maze, allowing you to reach the end without going through all the paths.<\/p>\n<p>For example, to evaluate [latex] f(x) = 6x^4 - x^3 - 15x^2 + 2x - 7 [\/latex] at [latex] x = 2 [\/latex], you can use synthetic division or directly substitute [latex] x [\/latex] with [latex]2[\/latex] to find the remainder.<\/p>\n<\/div>\n<section class=\"textbox example\">\n<p>Use the remainder theorem to evaluate [latex]f(x)=2x^5\u22123x^4\u22129x^3+8x^2+2 [\/latex] at [latex]x = -3[\/latex].<\/p>\n\n[reveal-answer q=\"981346\"]Show Solution[\/reveal-answer] [hidden-answer a=\"981346\"]\n\n<p>To find the remainder using the Remainder Theorem, use synthetic division to divide the polynomial by [latex]x + 3[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex] \\begin{array}{c|cccccc} -3 &amp; 2 &amp; -3 &amp; -9 &amp; 8 &amp; 0 &amp; 2 \\\\ &amp; &amp; -6 &amp; 27 &amp; -54 &amp; 138 &amp; -414 \\\\ \\hline &amp; 2 &amp; -9 &amp; 18 &amp; -46 &amp; 138 &amp; -412 \\\\ \\end{array} [\/latex]<\/p>\n<p>The remainder is [latex]-412[\/latex]. Therefore, [latex]f(-3) = -412[\/latex].<\/p>\n<strong>Analysis<\/strong>\n<p>We can check our answer by evaluating [latex]f(-3)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} f(x) &amp; = &amp; 2x^5 - 3x^4 - 9x^3 + 8x^2 + 2 \\\\ f(-3) &amp; =&nbsp; &amp;2(-3)^5 - 3(-3)^4 - 9(-3)^3 + 8(-3)^2 + 2 \\\\ &amp; = &amp; -412 \\end{array}[\/latex]<\/p>\n\n[\/hidden-answer]<\/section>\n<p>Watch the following video for more on the remainder theorem.<\/p>\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328534&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=p1lSRAeEMR0&amp;video_target=tpm-plugin-8u5s4j9s-p1lSRAeEMR0\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Remainder+Theorem+and+Synthetic+Division+of+Polynomials.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cRemainder Theorem and Synthetic Division of Polynomials\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Using the Factor Theorem to Solve a Polynomial Equation<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>The Factor Theorem is a bridge between zeros and factors of a polynomial. It states that a number [latex] k [\/latex] is a zero of [latex] f(x) [\/latex] if and only if [latex] (x - k) [\/latex] is a factor of [latex] f(x) [\/latex]. This theorem is like a two-way street; knowing a zero lets you find a factor, and knowing a factor lets you find a zero.<\/p>\n<p>For instance, to show that [latex] (x + 2) [\/latex] is a factor of [latex] x^3 - 6x^2 - x + 30 [\/latex], you can use synthetic division. If the remainder is zero, then [latex] (x + 2) [\/latex] is indeed a factor, and you can further factorize the quotient to find the remaining zeros of the polynomial.<\/p>\n<\/div>\n<section class=\"textbox example\">\n<p>Use the Factor Theorem to find the zeros of [latex]f(x)=x^3+4x^2\u22124x\u221216[\/latex] given that [latex](x\u22122)[\/latex] is a factor of the polynomial.<\/p>\n\n[reveal-answer q=\"981336\"]Show Solution[\/reveal-answer] [hidden-answer a=\"981336\"]\n\n<p>We can use synthetic division to show that [latex](x -2)[\/latex] is a factor of the polynomial.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c|cccc} 2 &amp; 1 &amp; 4 &amp; -4 &amp; -16 \\\\ &amp; &amp; 2 &amp; 12 &amp; 16 \\\\ \\hline &amp; 1 &amp; 6 &amp; 8 &amp; 0 \\\\ \\end{array}[\/latex]<\/p>\n<p>The remainder is zero, indicating that [latex](x-2)[\/latex] is indeed a factor of the polynomial. Using the Division Algorithm, we can express the polynomial as a product of the divisor and the quotient:<\/p>\n<p style=\"text-align: center;\">[latex] (x-2)(x^2 + 6x + 8) [\/latex]<\/p>\n<p>Now, let's further factor the quadratic polynomial [latex]x^2 + 6x + 8[\/latex].<\/p>\n<p>Using the middle term factorization method, the factors can be derived as:<\/p>\n<p style=\"text-align: center;\">[latex]x^2 + 4x + 2x + 8[\/latex]<\/p>\n<p>which gives:<\/p>\n<p style=\"text-align: center;\">[latex]x(x + 4) + 2(x + 4)[\/latex]<\/p>\n<p>resulting in:<\/p>\n<p style=\"text-align: center;\">[latex](x + 4)(x + 2)[\/latex]<\/p>\n<p>Combining with the previously found factor, we get:<\/p>\n<p style=\"text-align: center;\">[latex](x-2)(x + 4)(x + 2)[\/latex]<\/p>\n<p>Concluding from the Factor Theorem, the zeros of [latex]f(x) = x^3 + 4x^2 - 4x - 16[\/latex] are [latex]x = 2[\/latex], [latex]x = -2[\/latex], and [latex]x = -4[\/latex].<\/p>\n\n[\/hidden-answer]<\/section>\n<h2>Zeros of Polynomial Functions<\/h2>\n<p>When we talk about the zeros of polynomial functions, we're essentially looking for the values that make the polynomial equal to zero. These are the points where the graph of the polynomial will intersect the [latex]x[\/latex]-axis. It's like finding the roots of a tree that are hidden underground; they're essential for the structure but not immediately visible.<\/p>\n<h3>Using the Rational Zero Theorem to Find Rational Zeros<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>When we're navigating the complex sea of polynomial functions, the Rational Zero Theorem is like our compass. It helps us pinpoint potential rational zeros using a simple yet powerful strategy: looking at the factors of the constant term and the leading coefficient.<\/p>\n<p>For instance, consider a polynomial function that has zeros at [latex]\\frac{2}{5}[\/latex] and [latex]\\frac{3}{4}[\/latex]. These aren't just numbers; they're clues. By setting up equations with these zeros and constructing a quadratic function, we can see a pattern emerge. The numerators of these zeros ([latex]2[\/latex] and [latex]3[\/latex]) are factors of the constant term, while the denominators ([latex]5[\/latex] and [latex]4[\/latex]) are factors of the leading coefficient.<\/p>\n<br>\n<strong>Quick Tips: Applying the Rational Zero Theorem<\/strong>\n<ol style=\"list-style-type: decimal;\">\n\t<li><strong>List Out Factors<\/strong>: Start by listing all factors of the constant term and the leading coefficient of your polynomial.<\/li>\n\t<li><strong>Form Ratios<\/strong>: Create all possible ratios [latex]\\frac{p}{q}[\/latex] where [latex]p[\/latex] is a factor of the constant term and [latex]q[\/latex] is a factor of the leading coefficient.<\/li>\n\t<li><strong>Test Your Candidates<\/strong>: Evaluate each potential zero by plugging it into the polynomial. If the result is zero, you've found a true zero!<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">\n<p>Use the Rational Zero Theorem to find the rational zeros of [latex]f(x) = 2x^3 + x^2 - 4x + 1[\/latex].<\/p>\n\n[reveal-answer q=\"981243\"]Show Solution[\/reveal-answer] [hidden-answer a=\"981243\"]\n\n<p>The Rational Zero Theorem tells us that if [latex]\\frac{p}{q}[\/latex] is a zero of [latex]f(x)[\/latex], then [latex]p[\/latex] is a factor of [latex]1[\/latex] and [latex]q[\/latex] is a factor of [latex]2[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c c} \\frac{p}{q} &amp; = \\frac{\\text{factor of constant term}}{\\text{factor of leading coefficient}} \\\\ &amp; = \\frac{\\text{factor of 1}}{\\text{factor of 2}} \\end{array} [\/latex]<\/p>\n<p>The factors of [latex]1[\/latex] are [latex]\\pm1[\/latex] and the factors of [latex]2[\/latex] are [latex]\\pm1[\/latex] and [latex]\\pm2[\/latex]. The possible values for [latex]\\frac{p}{q}[\/latex] are [latex]\\pm1[\/latex] and [latex]\\pm\\frac{1}{2}[\/latex]. These are the possible rational zeros for the function. We can determine which of the possible zeros are actual zeros by substituting these values for [latex]x[\/latex] in [latex]f(x)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f(-1) = 2(-1)^3 + (-1)^2 - 4(-1) + 1 = 4[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]f(1) = 2(1)^3 + (1)^2 - 4(1) + 1 = 0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(-\\frac{1}{2}\\right) = 2\\left(-\\frac{1}{2}\\right)^3 + \\left(-\\frac{1}{2}\\right)^2 - 4\\left(-\\frac{1}{2}\\right) + 1 = 3[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(\\frac{1}{2}\\right) = 2\\left(\\frac{1}{2}\\right)^3 + \\left(\\frac{1}{2}\\right)^2 - 4\\left(\\frac{1}{2}\\right) + 1 = -\\frac{1}{2}[\/latex]<\/p>\n<p>Of those, [latex]-1, -\\frac{1}{2},[\/latex] and [latex]\\frac{1}{2}[\/latex] are not zeros of [latex]f(x)[\/latex]. [latex]1[\/latex] is the only rational zero of [latex]f(x)[\/latex].<\/p>\n\n[\/hidden-answer]<\/section>\n<section class=\"textbox example\">\n<p>Use the Rational Zero Theorem to find the rational zeros of [latex]f(x)=x^3\u22125x^2+2x+1[\/latex].<\/p>\n\n[reveal-answer q=\"981143\"]Show Solution[\/reveal-answer] [hidden-answer a=\"981143\"]\n\n<p>There are no rational zeros.<\/p>\n\n[\/hidden-answer]<\/section>\n<p>Watch the following video see more on the rational zero theorem.<\/p>\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328535&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=Iaq7z7reznM&amp;video_target=tpm-plugin-3vmzk5cw-Iaq7z7reznM\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Finding+All+Zeros+of+a+Polynomial+Function+Using+The+Rational+Zero+Theorem.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding All Zeros of a Polynomial Function Using The Rational Zero Theorem\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Graphs of Polynomial Functions<\/h2>\n<h3>Zeros and Multiplicity<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>The zeros of a polynomial function are the points where the graph crosses the x-axis, meaning the output [latex]y[\/latex] is zero. For example, the polynomial function [latex]f(x)=(x+3)(x\u22122)^2(x+1)^3[\/latex]has zeros at:<\/p>\n<ul>\n\t<li>[latex]x = -3[\/latex]<\/li>\n\t<li>[latex]x = 2[\/latex]<\/li>\n\t<li>[latex]x = -1[\/latex]<\/li>\n<\/ul>\n<strong>Quick Tips: Multiplicity and Graph Behavior<\/strong>\n<ul>\n\t<li>Single Zeros: When a factor appears once, such as [latex]x+3[\/latex], the graph crosses the [latex]x[\/latex]-axis at this zero. This is a zero with multiplicity [latex]1[\/latex].<\/li>\n\t<li>Even Multiplicity: Zeros with an even multiplicity, like [latex]x=2[\/latex] in [latex](x\u22122)^2[\/latex], indicate the graph touches the x-axis and turns around.<\/li>\n\t<li>Odd Multiplicity: Zeros with an odd multiplicity, such as [latex]x=\u22121[\/latex] in [latex](x+1)^3[\/latex], show that the graph will cross the [latex]x[\/latex]-axis.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Use the graph of the function of degree [latex]5[\/latex] to identify the zeros of the function and their multiplicities.<\/p>\n<center><img class=\"small aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201600\/CNX_Precalc_Figure_03_04_0102.jpg\" alt=\"Graph of an even-degree polynomial with degree 6.\" width=\"487\" height=\"253\"><\/center>\n<p>&nbsp;<\/p>\n<p>[reveal-answer q=\"874458\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"874458\"]<\/p>\n<p>The graph has a zero of [latex]\u20135[\/latex] with multiplicity [latex]1[\/latex], a zero of [latex]\u20131[\/latex] with multiplicity [latex]2[\/latex], and a zero of [latex]3[\/latex] with multiplicity [latex]2[\/latex].<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Use the graph below to find the zeros of the degree [latex]6[\/latex] function and their multiplicities.<\/p>\n<p>&nbsp;<\/p>\n<center><img class=\"aligncenter wp-image-6767\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/11\/08231949\/Screen-Shot-2019-07-08-at-4.19.09-PM.png\" alt=\"Graph of a polynomial that goes through\/touches the x-axis at (-3,0), (-2,0), and (1,0)\" width=\"503\" height=\"494\"><\/center>\n<p>&nbsp;<\/p>\n<p>[reveal-answer q=\"953557\"]Show Solution[\/reveal-answer]<\/p>\n<p>[hidden-answer a=\"953557\"]<\/p>\n<table>\n<tbody>\n<tr>\n<td>Zero<\/td>\n<td>Multiplicity<\/td>\n<\/tr>\n<tr>\n<td>[latex](-3,0)[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex](-2,0)[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex](1,0)[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<p>Watch the following video see an example of finding the zeros and their multiplicities.<\/p>\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328536&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=VP_7Zy1YDE4&amp;video_target=tpm-plugin-k2nntq0b-VP_7Zy1YDE4\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/How+to+Determine+the+Multiplicity+and+Zeros+of+a+Polynomial.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Determine the Multiplicity and Zeros of a Polynomial\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Determining the Number of Turning Points and Intercepts from the Degree of the Polynomial<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>The number of turning points in a polynomial graph is related to the degree of the polynomial. A polynomial of degree [latex]n[\/latex] can have up to [latex]n\u22121[\/latex] turning points. For instance, a [latex]4[\/latex]th-degree polynomial can have up to [latex]3[\/latex] turning points. This is an essential concept in visualizing the shape of the graph.<\/p>\n\nThe degree and leading coefficient of a polynomial function influence its end behavior. For a polynomial [latex]f(x)=ax^n[\/latex]:\n\n<ul>\n\t<li>If [latex]n[\/latex] is odd and [latex]a&gt;0[\/latex], the ends of the graph go in opposite directions, with the right end going up.<\/li>\n\t<li>If [latex]n[\/latex] is even and [latex]a&gt;0[\/latex], both ends of the graph go up.<\/li>\n<\/ul>\n\nThis knowledge helps in sketching a rough graph of the polynomial function.<\/div>\n<section class=\"textbox example\">\n<p>Without graphing the function, determine the maximum number of <em>x<\/em>-intercepts and turning points for [latex]f\\left(x\\right)=108 - 13{x}^{9}-8{x}^{4}+14{x}^{12}+2{x}^{3}[\/latex].<\/p>\n<p>[reveal-answer q=\"304362\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"304362\"]<\/p>\n<p>There are at most [latex]12[\/latex] [latex]x[\/latex]-intercepts and at most [latex]11[\/latex] turning points.[\/hidden-answer]<\/p>\n<\/section>\n<p>The following video gives a 5 minute lesson on how to determine the number of intercepts and turning points of a polynomial function given its degree.<\/p>\n<section class=\"textbox watchIt\">\n<p>[embed]https:\/\/youtu.be\/9WW0EetLD4Q[\/embed]<\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Turning+Points+and+X+Intercepts+of+a+Polynomial+Function.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cTurning Points and X Intercepts of a Polynomial Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>&nbsp;<\/h2>\n<h3>Identifying&nbsp;the Shape of the Graph of a Polynomial Function<\/h3>\n<p>Polynomial functions are like the diverse landscapes of mathematics, where each function can represent a unique terrain. The graphs of these functions are smooth and continuous, meaning they have no sharp turns or breaks. Just like a rolling hillside, these functions have a fluidity that's both predictable and elegant.<\/p>\n<section class=\"textbox example\">\n<p>Which of the graphs below&nbsp;represents a polynomial function?<\/p>\n<p><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201420\/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>[reveal-answer q=\"207827\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"207827\"]<\/p>\n<p id=\"fs-id1165134129608\">The graphs of [latex]f[\/latex]&nbsp;and [latex]h[\/latex] are graphs of polynomial functions. They are smooth and <strong>continuous<\/strong>.<\/p>\n<p id=\"fs-id1165134188794\">The graphs of [latex]g[\/latex]&nbsp;and [latex]k[\/latex] are graphs of functions that are not polynomials. The graph of function [latex]g[\/latex]&nbsp;has a sharp corner. The graph of function [latex]k[\/latex]&nbsp;is not continuous.<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<h4>Identifying End Behavior of Polynomial Functions<\/h4>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<strong>Quick Tips: Even vs. Odd Degree Polynomials<\/strong>\n<ul>\n\t<li><strong>Even Degree Polynomials<\/strong>: Think of even-degree polynomials like [latex]f(x)=x^2, g(x)=x^4,[\/latex] and [latex]h(x)=x^6[\/latex] as the serene hills that either rise or fall on both ends. As the degree increases, the middle part of the graph becomes flatter, and the ends become steeper, much like a valley becoming more pronounced with time.<\/li>\n\t<li><strong>Odd Degree Polynomials<\/strong>: Odd-degree polynomials such as [latex]f(x)=x^3, g(x)=x^5,[\/latex] and [latex]h(x)=x^7[\/latex] are the adventurous mountain ranges of the polynomial world. One side ascends to the sky, while the other descends into the depths of the ocean. This is due to the fact that a negative input will yield a negative output when the degree is odd, creating a graph that has a perfect balance of ascent and descent.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Describe the end behavior of the polynomial function in the graph below.<\/p>\n<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194522\/CNX_Precalc_Figure_03_03_016n2.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"440\"><\/center>\n<p>&nbsp;<\/p>\n<p>[reveal-answer q=\"304329\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"304329\"]<\/p>\n<p>As [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty[\/latex] and as [latex]x\\to -\\infty , f\\left(x\\right)\\to -\\infty [\/latex]. It has the shape of an even degree power function with a negative coefficient.<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Given the function [latex]f\\left(x\\right)=0.2\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 5\\right)[\/latex], express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.<\/p>\n<p>[reveal-answer q=\"657153\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"657153\"]<\/p>\n<p>The leading term is [latex]0.2{x}^{3}[\/latex], so it is a degree [latex]3[\/latex] polynomial. As [latex]x[\/latex] approaches positive infinity, [latex]f\\left(x\\right)[\/latex] increases without bound; as [latex]x[\/latex] approaches negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound.[\/hidden-answer]<\/p>\n<\/section>\n<p>In the following video, you'll see more examples that summarize the end behavior of polynomial functions and which components of the function contribute to it.<\/p>\n<section class=\"textbox watchIt\">\n<p>https:\/\/youtu.be\/y78Dpr9LLN0<\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Summary+of+End+Behavior+or+Long+Run+Behavior+of+Polynomial+Functions.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSummary of End Behavior or Long Run Behavior of Polynomial Functions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h4>The Sign of the Leading Term<\/h4>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n\nThe leading term of a polynomial function is like the compass that dictates the direction of your graph. Changing the sign of this term is akin to entering a mirror world, where everything is flipped. <strong>Quick Tips: Even vs. Odd Degree Polynomials<\/strong>\n<ul>\n\t<li>Positive Leading Term: When the leading term is positive, as in 3x^4[\/latex], the arms of the graph reach for the heavens. The graph stays positive, no matter if [latex]x[\/latex] is negative or positive.<\/li>\n\t<li>Negative Leading Term: Introduce a negative sign, and the graph of [latex]\u22123x^4[\/latex] turns upside down, diving down as if reflecting in a lake. The negative leading term ensures that the arms of the polynomial are pointing downwards, regardless of the value of [latex]x[\/latex].<\/li>\n<\/ul>\n<\/div>\n<p>The table below summarizes all four cases mentioned.<\/p>\n<table>\n<thead>\n<tr>\n<th style=\"text-align: center;\">Even Degree<\/th>\n<th style=\"text-align: center;\">Odd Degree<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/11.png\"><img class=\"alignnone size-full wp-image-12504\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201602\/11.png\" alt=\"11\" width=\"423\" height=\"559\"><\/a><\/td>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/12.png\"><img class=\"alignnone size-full wp-image-12505\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201605\/12.png\" alt=\"12\" width=\"397\" height=\"560\"><\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/13.png\"><img class=\"alignnone size-full wp-image-12506\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201607\/13.png\" alt=\"13\" width=\"387\" height=\"574\"><\/a><\/td>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/14.png\"><img class=\"alignnone size-full wp-image-12507\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201609\/14.png\" alt=\"14\" width=\"404\" height=\"564\"><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Sketching Graphs<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<br>\n<br>\n<\/strong>When we sketch the graphs of polynomial functions, we're essentially drawing a rough map of how the function behaves. It's like sketching the path of a hiker in the mountains: we mark the starting point, the peaks, and the valleys. Here's how we do it:<\/p>\n<ul>\n\t<li><strong>Intercepts<\/strong>: These are our starting points. Just like a hiker checks a map for where to begin, we find where our graph crosses the axes.<\/li>\n\t<li><strong>Symmetry<\/strong>: Is our path mirrored on either side? If our function is even, the graph is symmetric like butterfly wings across the [latex]y[\/latex]-axis ([latex]f(\u2212x)=f(x)[\/latex]). If it's odd, it's like a twisty mountain road that flips across the origin ([latex]f(\u2212x)=\u2212f(x)[\/latex]).<\/li>\n\t<li><strong>Multiplicities and Intercepts<\/strong>: This tells us how our hiker approaches each peak or valley. Does she pass straight through, or does she hesitate and turn back a bit? The multiplicity of zeros gives us this behavior at [latex]x[\/latex]-intercepts.<\/li>\n\t<li><strong>End Behavior<\/strong>: This is the direction our hiker heads off into the distance. It's determined by the leading term of our polynomial.<\/li>\n\t<li><strong>Sketching the Graph<\/strong>: With all this information, we can draw a rough trail of our function, ensuring we don't have more twists and turns (turning points) than the degree of the polynomial allows.<\/li>\n\t<li><strong>Technology Check<\/strong>: Just like a hiker might check GPS, we can use graphing technology to confirm our path.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">Sketch a possible graph for [latex]f\\left(x\\right)=\\frac{1}{4}x{\\left(x - 1\\right)}^{4}{\\left(x+3\\right)}^{3}[\/latex]. Check yourself with an online graphing calculator when you are done. [reveal-answer q=\"812296\"]Show Solution[\/reveal-answer] [hidden-answer a=\"812296\"]<center><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/30233150\/CNX_Precalc_Figure_03_04_0212.jpg\"><img class=\"aligncenter size-full wp-image-2911\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/30233150\/CNX_Precalc_Figure_03_04_0212.jpg\" alt=\"Graph of f(x)=(1\/4)x(x-1)^4(x+3)^3.\" width=\"487\" height=\"366\"><\/a><\/center>[\/hidden-answer]<\/section>\n<p>Watch the following video to see an example of sketching a polynomial.<\/p>\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328600&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=PWLvZZfpC5I&amp;video_target=tpm-plugin-g3fbwvu9-PWLvZZfpC5I\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/How+to+Sketch+a+Polynomial+Function.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Sketch a Polynomial Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>The following video is a bit long, but it covers how to graph a polynomial function using end behavior, multiplicity, and zeros.<\/p>\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328601&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=a5x4lwnvHM0&amp;video_target=tpm-plugin-2oau5nr7-a5x4lwnvHM0\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/How+To+Graph+Polynomial+Functions+Using+End+Behavior%2C+Multiplicity+%26+Zeros.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow To Graph Polynomial Functions Using End Behavior, Multiplicity &amp; Zeros\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Writing Formulas for Polynomial Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<br>\n<br>\n<\/strong>Writing the formula for a polynomial function from its graph is like retracing the hiker's steps to map her exact route:<\/p>\n<ul>\n\t<li><strong>[latex]X[\/latex]-Intercepts<\/strong>: Find where the hiker started and finished her journey. These give us the factors of the polynomial.<\/li>\n\t<li><strong>Multiplicity<\/strong>: How did she approach each peak? Did she go straight over, or did she pause and reflect? This tells us the multiplicity of each factor.<\/li>\n\t<li><strong>Least Degree<\/strong>: We want the simplest map that still shows all the important points. Find the polynomial of the least degree that includes all factors.<\/li>\n\t<li><strong>Stretch Factor<\/strong>: How steep were the mountains? Use another point (like the [latex]y[\/latex]-intercept) to figure out the \"stretch\" of our polynomial, which scales the function up or down.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Given the graph below, write a formula for the function shown.<\/p>\n<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201629\/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\"><\/center>[reveal-answer q=\"427364\"]Show Solution[\/reveal-answer] [hidden-answer a=\"427364\"]\n\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]&nbsp;<\/p>\n\n[\/hidden-answer]<\/section>\n<p>Watch the following video to see an example of writing the equation of a polynomial.<\/p>\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/m9PM6Z2b55A?si=4QJXnusXTgfdJ86w\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex1_+Find+an+Equation+of+a+Degree+4+Polynomial+Function+From+the+Graph+of+the+Function.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx1: Find an Equation of a Degree 4 Polynomial Function From the Graph of the Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Recognize polynomial functions, noting their degree and leading coefficient<\/li>\n<li>Apply various methods to divide polynomials and locate the zeros of polynomial equations<\/li>\n<li>Generate graphs and formulate equations for polynomial functions<\/li>\n<li>Explain the end behavior of polynomial functions<\/li>\n<\/ul>\n<\/section>\n<h2>Identifying Polynomial Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>Polynomial functions are like the DNA of algebra\u2014they combine simplicity and complexity to form an incredibly diverse array of functions.<\/p>\n<p>At their core, polynomials are sums of terms made up of coefficients and variables raised to whole number powers. The degree of a polynomial, given by the highest power of the variable, tells us a lot about the function&#8217;s behavior and the shape of its graph.<\/p>\n<p>Let [latex]n[\/latex]&nbsp; be a non-negative integer. A <strong>polynomial function<\/strong> is a function that can be written in the form<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/p>\n<p>This is called the <strong>general form of a polynomial function<\/strong>.<\/p>\n<p>Each [latex]{a}_{i}[\/latex]&nbsp;is a coefficient and can be any real number.<\/p>\n<p>Each product [latex]{a}_{i}{x}^{i}[\/latex]&nbsp;is a <strong>term of a polynomial function<\/strong>.<\/p>\n<p><strong>Quick Tips<\/strong><\/p>\n<ul>\n<li><strong>Linear, Quadratic, and Beyond:<\/strong> A first-degree polynomial is linear, a second-degree is quadratic, and higher degrees have their own characteristics and complexities.<\/li>\n<li><strong>Coefficient Clues:<\/strong> The coefficients in a polynomial can tell us about the steepness and direction of the graph. A positive leading coefficient means the graph opens upward, and a negative one indicates it opens downward.<\/li>\n<\/ul>\n<\/div>\n<h2>Defining the Degree and Leading Coefficient of a Polynomial Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>The degree of a polynomial is the highest power of the variable present. The leading coefficient is the coefficient of the term with the highest power. These two characteristics can tell us a lot about the function&#8217;s behavior, especially its growth and end behavior.<\/p>\n<ul>\n<li>\n<p><strong>Degree Tells the Tale:<\/strong> The degree of a polynomial function hints at the number of roots and the possible number of turns on its graph. For instance, a second-degree polynomial, or a quadratic, will have at most two roots and one turn.<\/p>\n<\/li>\n<li>\n<p><strong>Leading Coefficient Impact:<\/strong> The leading coefficient isn&#8217;t just a number\u2014it&#8217;s the boss. It influences the end behavior of the polynomial&#8217;s graph. If it&#8217;s positive, the graph eventually rises; if negative, the graph falls.<\/p>\n<\/li>\n<\/ul>\n<p><strong>Quick Tips: Identifying Degree and Leading Coefficient<\/strong><\/p>\n<ul>\n<li>Arrange terms in descending order of power to easily identify the leading term and coefficient.<\/li>\n<li>The degree gives a sense of the &#8216;shape&#8217; of the graph of the polynomial function.<\/li>\n<li>For instance, [latex]f(x)=3+2x^2\u22124x^3[\/latex] has a degree of [latex]3[\/latex] and a leading coefficient of [latex]-4[\/latex], indicating a cubic function with a negative leading term.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Identify the degree, leading term, and leading coefficient of the polynomial [latex]f\\left(x\\right)=4{x}^{2}-{x}^{6}+2x - 6[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q435637\">Show Solution<\/button><\/p>\n<div id=\"q435637\" class=\"hidden-answer\" style=\"display: none\">\n<p>The degree is [latex]6[\/latex]. The leading term is [latex]-{x}^{6}[\/latex]. The leading coefficient is [latex]\u20131[\/latex].<\/p><\/div>\n<\/div>\n<\/section>\n<p>Watch the following video for more examples of how to determine the degree, leading term, and leading coefficient of a polynomial.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Degree, Leading Term, and Leading Coefficient of a Polynomial Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/F_G_w82s0QA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Degree%2C+Leading+Term%2C+and+Leading+Coefficient+of+a+Polynomial+Function.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDegree, Leading Term, and Leading Coefficient of a Polynomial Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Polynomial Long Division<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>Polynomial long division might seem daunting at first, but it&#8217;s quite similar to the long division we learned in elementary school. Just like we divide numbers, we can divide polynomials, breaking them down into simpler parts. The process involves dividing the highest degree terms and then subtracting to find the remainder, which we then bring down the next term to continue the process.<\/p>\n<p><strong>Quick Tips: Steps to Polynomial Long Division<\/strong><\/p>\n<ol>\n<li><strong>Set Up<\/strong>: Write the polynomial (dividend) and the binomial (divisor) in long division format.<\/li>\n<li><strong>Divide the Leading Terms<\/strong>: Take the leading term of the dividend and divide it by the leading term of the divisor to find the first term of the quotient.<\/li>\n<li><strong>Multiply and Subtract<\/strong>: Multiply the entire divisor by the term just found and subtract it from the dividend.<\/li>\n<li><strong>Repeat<\/strong>: Bring down the next term of the dividend and repeat the process until you&#8217;ve worked through each term.<\/li>\n<li><strong>Remainder<\/strong>: If there&#8217;s a remainder, express it as a fraction with the divisor as the denominator.<\/li>\n<\/ol>\n<p><strong>The Division Algorithm: A Formal Approach<\/strong><\/p>\n<p>The Division Algorithm is a fancy way of saying that any polynomial divided by another can be expressed as a multiplication (the quotient) plus something left over (the remainder). It&#8217;s a structured method to ensure that every polynomial division can be broken down neatly.<\/p>\n<\/div>\n<section class=\"textbox example\">\n<p>Divide [latex]5{x}^{2}+3x - 2[\/latex] by [latex]x+1[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q996959\">Show Solution<\/button><\/p>\n<div id=\"q996959\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204327\/CNX_Precalc_revised_eq_22.png\" alt=\"Set up the division problem. 5x squared divided by x is 5x. Multiply x plus 1 by 5x. Subtract. Bring down the next term. Negative 2x divded by x is negative 2. Multiply x + 1 by negative 2. Subtract.\" width=\"426\" height=\"288\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>The quotient is [latex]5x - 2[\/latex].&nbsp;The remainder is [latex]0[\/latex]. We write the result as<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{5{x}^{2}+3x - 2}{x+1}=5x - 2[\/latex]<\/p>\n<p>or<\/p>\n<p style=\"text-align: center;\">[latex]5{x}^{2}+3x - 2=\\left(x+1\\right)\\left(5x - 2\\right)[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>This division problem had a remainder of [latex]0[\/latex]. This tells us that the dividend is divided evenly by the divisor and that the divisor is a factor of the dividend.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Divide [latex]16{x}^{3}-12{x}^{2}+20x - 3[\/latex]&nbsp;by [latex]4x+5[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q198989\">Show Solution<\/button><\/p>\n<div id=\"q198989\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]4{x}^{2}-8x+15-\\frac{78}{4x+5}[\/latex]<\/p><\/div>\n<\/div>\n<\/section>\n<p>Watch the following video see an example of dividing polynomials with long division.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328532&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=U2f5VzKGNPk&amp;video_target=tpm-plugin-yrwb7tah-U2f5VzKGNPk\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Dividing+polynomials+using+long+division.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDividing polynomials using long division\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Using Synthetic Division to Divide Polynomials<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>Synthetic division is a quicker way to divide polynomials when the divisor is a binomial with a leading coefficient of [latex]1[\/latex]. It uses only the coefficients of the polynomials, simplifying the process significantly.<\/p>\n<p><strong>How to Use Synthetic Division<\/strong><\/p>\n<ol>\n<li><strong>Write the Opposite of k<\/strong>: If dividing by [latex]x-k[\/latex], write down [latex]k[\/latex].<\/li>\n<li><strong>List the Coefficients<\/strong>: Write the coefficients of the polynomial you&#8217;re dividing.<\/li>\n<li><strong>Bring Down the Leading Coefficient<\/strong>: This starts off your synthetic division.<\/li>\n<li><strong>Multiply and Add<\/strong>: Multiply the leading coefficient by [latex]k[\/latex], add it to the next coefficient, and continue this process.<\/li>\n<li><strong>Find the Quotient and Remainder<\/strong>: The last number is the remainder, and the others form the coefficients of the quotient polynomial.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Use synthetic division to divide [latex]5x^2 - 3x - 36[\/latex] by [latex]x - 3[\/latex]. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q981343\">Show Solution<\/button> <\/p>\n<div id=\"q981343\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by setting up the synthetic division. Write [latex]k[\/latex] and the coefficients.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-9522 size-full\" src=\"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-content\/uploads\/sites\/18\/2023\/10\/9f9b7c3f2afaf1c98578471f8ec6fd5c3b34749f.webp#fixme\" alt=\"A collapsed version of the previous synthetic division.\" width=\"487\" height=\"55\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>Bring down the lead coefficient. Multiply the lead coefficient by [latex]k[\/latex].<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-9523 size-full\" src=\"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-content\/uploads\/sites\/18\/2023\/10\/8b638e9a3fafd6b1e6496583a605b4a0b4858d0e.webp#fixme\" alt=\"The set-up of the synthetic division for the polynomial 5x^2-3x-36 by x-3, which renders {5, -3, -36} by 3.\" width=\"487\" height=\"74\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>Continue by adding the numbers in the second column. Multiply the resulting number by [latex]k[\/latex]. Write the result in the next column. Then add the numbers in the third column.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-9524 size-full\" src=\"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-content\/uploads\/sites\/18\/2023\/10\/a77e7bc9a6d09d92cfe1e1f485fc8887a5ba4e6f.webp#fixme\" alt=\"Multiplied by the lead coefficient, 5, in the second column, and the lead coefficient is brought down to the second row.\" width=\"487\" height=\"74\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>The result is [latex]5x + 12[\/latex]. The remainder is [latex]0[\/latex]. So [latex]x - 3[\/latex] is a factor of the original polynomial.<\/p>\n<p><strong>Analysis<\/strong><\/p>\n<p>Just as with long division, we can check our work by multiplying the quotient by the divisor and adding the remainder.<\/p>\n<div style=\"text-align: center;\">[latex](x\u22123)(5x+12)+0=5x^2\u22123x\u221236[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Use synthetic division to divide [latex]-9x^4 + 10x^3 + 7x^2 - 6[\/latex] by [latex]x - 1[\/latex]. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q981344\">Show Solution<\/button> <\/p>\n<div id=\"q981344\" class=\"hidden-answer\" style=\"display: none\">\n<p>Notice there is no [latex]x[\/latex]-term. We will use a zero as the coefficient for that term.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-9533\" src=\"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-content\/uploads\/sites\/18\/2023\/09\/21c33f8114c651702043fec1fafe25f629656a9a.webp#fixme\" alt=\"A synthetic division table showing the division of a polynomial by the binomial (x - 1). The divisor '1' is on the far left. The row at the top lists the coefficients of the polynomial: -9, 10, 7, 0, and -6. The second row represents the products of the divisor and the results of each step: -9, 1, 8, and 8. The bottom row sums the coefficients and the products: -9, 1, 8, 8, and 2.\" width=\"205\" height=\"76\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>The result is [latex]-9x^3 + x^2 + 8x + 8 + \\frac{2}{x-1}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video see an example of dividing polynomials using synthetic division.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328533&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=FxHWoUOq2iQ&amp;video_target=tpm-plugin-6i7czt2l-FxHWoUOq2iQ\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Synthetic+Division+of+Polynomials.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSynthetic Division of Polynomials\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Evaluate Polynomials<\/h2>\n<h3>Evaluating a Polynomial Using the Remainder Theorem<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>The Remainder Theorem is a quick way to evaluate polynomials. If you divide a polynomial [latex]f(x)[\/latex]by [latex]x - k[\/latex], the remainder is simply the value of [latex]f(k)[\/latex]. It&#8217;s like a shortcut in a maze, allowing you to reach the end without going through all the paths.<\/p>\n<p>For example, to evaluate [latex]f(x) = 6x^4 - x^3 - 15x^2 + 2x - 7[\/latex] at [latex]x = 2[\/latex], you can use synthetic division or directly substitute [latex]x[\/latex] with [latex]2[\/latex] to find the remainder.<\/p>\n<\/div>\n<section class=\"textbox example\">\n<p>Use the remainder theorem to evaluate [latex]f(x)=2x^5\u22123x^4\u22129x^3+8x^2+2[\/latex] at [latex]x = -3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q981346\">Show Solution<\/button> <\/p>\n<div id=\"q981346\" class=\"hidden-answer\" style=\"display: none\">\n<p>To find the remainder using the Remainder Theorem, use synthetic division to divide the polynomial by [latex]x + 3[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c|cccccc} -3 & 2 & -3 & -9 & 8 & 0 & 2 \\\\ & & -6 & 27 & -54 & 138 & -414 \\\\ \\hline & 2 & -9 & 18 & -46 & 138 & -412 \\\\ \\end{array}[\/latex]<\/p>\n<p>The remainder is [latex]-412[\/latex]. Therefore, [latex]f(-3) = -412[\/latex].<\/p>\n<p><strong>Analysis<\/strong><\/p>\n<p>We can check our answer by evaluating [latex]f(-3)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} f(x) & = & 2x^5 - 3x^4 - 9x^3 + 8x^2 + 2 \\\\ f(-3) & =&nbsp; &2(-3)^5 - 3(-3)^4 - 9(-3)^3 + 8(-3)^2 + 2 \\\\ & = & -412 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video for more on the remainder theorem.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328534&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=p1lSRAeEMR0&amp;video_target=tpm-plugin-8u5s4j9s-p1lSRAeEMR0\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Remainder+Theorem+and+Synthetic+Division+of+Polynomials.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cRemainder Theorem and Synthetic Division of Polynomials\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Using the Factor Theorem to Solve a Polynomial Equation<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>The Factor Theorem is a bridge between zeros and factors of a polynomial. It states that a number [latex]k[\/latex] is a zero of [latex]f(x)[\/latex] if and only if [latex](x - k)[\/latex] is a factor of [latex]f(x)[\/latex]. This theorem is like a two-way street; knowing a zero lets you find a factor, and knowing a factor lets you find a zero.<\/p>\n<p>For instance, to show that [latex](x + 2)[\/latex] is a factor of [latex]x^3 - 6x^2 - x + 30[\/latex], you can use synthetic division. If the remainder is zero, then [latex](x + 2)[\/latex] is indeed a factor, and you can further factorize the quotient to find the remaining zeros of the polynomial.<\/p>\n<\/div>\n<section class=\"textbox example\">\n<p>Use the Factor Theorem to find the zeros of [latex]f(x)=x^3+4x^2\u22124x\u221216[\/latex] given that [latex](x\u22122)[\/latex] is a factor of the polynomial.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q981336\">Show Solution<\/button> <\/p>\n<div id=\"q981336\" class=\"hidden-answer\" style=\"display: none\">\n<p>We can use synthetic division to show that [latex](x -2)[\/latex] is a factor of the polynomial.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c|cccc} 2 & 1 & 4 & -4 & -16 \\\\ & & 2 & 12 & 16 \\\\ \\hline & 1 & 6 & 8 & 0 \\\\ \\end{array}[\/latex]<\/p>\n<p>The remainder is zero, indicating that [latex](x-2)[\/latex] is indeed a factor of the polynomial. Using the Division Algorithm, we can express the polynomial as a product of the divisor and the quotient:<\/p>\n<p style=\"text-align: center;\">[latex](x-2)(x^2 + 6x + 8)[\/latex]<\/p>\n<p>Now, let&#8217;s further factor the quadratic polynomial [latex]x^2 + 6x + 8[\/latex].<\/p>\n<p>Using the middle term factorization method, the factors can be derived as:<\/p>\n<p style=\"text-align: center;\">[latex]x^2 + 4x + 2x + 8[\/latex]<\/p>\n<p>which gives:<\/p>\n<p style=\"text-align: center;\">[latex]x(x + 4) + 2(x + 4)[\/latex]<\/p>\n<p>resulting in:<\/p>\n<p style=\"text-align: center;\">[latex](x + 4)(x + 2)[\/latex]<\/p>\n<p>Combining with the previously found factor, we get:<\/p>\n<p style=\"text-align: center;\">[latex](x-2)(x + 4)(x + 2)[\/latex]<\/p>\n<p>Concluding from the Factor Theorem, the zeros of [latex]f(x) = x^3 + 4x^2 - 4x - 16[\/latex] are [latex]x = 2[\/latex], [latex]x = -2[\/latex], and [latex]x = -4[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Zeros of Polynomial Functions<\/h2>\n<p>When we talk about the zeros of polynomial functions, we&#8217;re essentially looking for the values that make the polynomial equal to zero. These are the points where the graph of the polynomial will intersect the [latex]x[\/latex]-axis. It&#8217;s like finding the roots of a tree that are hidden underground; they&#8217;re essential for the structure but not immediately visible.<\/p>\n<h3>Using the Rational Zero Theorem to Find Rational Zeros<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>When we&#8217;re navigating the complex sea of polynomial functions, the Rational Zero Theorem is like our compass. It helps us pinpoint potential rational zeros using a simple yet powerful strategy: looking at the factors of the constant term and the leading coefficient.<\/p>\n<p>For instance, consider a polynomial function that has zeros at [latex]\\frac{2}{5}[\/latex] and [latex]\\frac{3}{4}[\/latex]. These aren&#8217;t just numbers; they&#8217;re clues. By setting up equations with these zeros and constructing a quadratic function, we can see a pattern emerge. The numerators of these zeros ([latex]2[\/latex] and [latex]3[\/latex]) are factors of the constant term, while the denominators ([latex]5[\/latex] and [latex]4[\/latex]) are factors of the leading coefficient.<\/p>\n<p>\n<strong>Quick Tips: Applying the Rational Zero Theorem<\/strong><\/p>\n<ol style=\"list-style-type: decimal;\">\n<li><strong>List Out Factors<\/strong>: Start by listing all factors of the constant term and the leading coefficient of your polynomial.<\/li>\n<li><strong>Form Ratios<\/strong>: Create all possible ratios [latex]\\frac{p}{q}[\/latex] where [latex]p[\/latex] is a factor of the constant term and [latex]q[\/latex] is a factor of the leading coefficient.<\/li>\n<li><strong>Test Your Candidates<\/strong>: Evaluate each potential zero by plugging it into the polynomial. If the result is zero, you&#8217;ve found a true zero!<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">\n<p>Use the Rational Zero Theorem to find the rational zeros of [latex]f(x) = 2x^3 + x^2 - 4x + 1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q981243\">Show Solution<\/button> <\/p>\n<div id=\"q981243\" class=\"hidden-answer\" style=\"display: none\">\n<p>The Rational Zero Theorem tells us that if [latex]\\frac{p}{q}[\/latex] is a zero of [latex]f(x)[\/latex], then [latex]p[\/latex] is a factor of [latex]1[\/latex] and [latex]q[\/latex] is a factor of [latex]2[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c c} \\frac{p}{q} & = \\frac{\\text{factor of constant term}}{\\text{factor of leading coefficient}} \\\\ & = \\frac{\\text{factor of 1}}{\\text{factor of 2}} \\end{array}[\/latex]<\/p>\n<p>The factors of [latex]1[\/latex] are [latex]\\pm1[\/latex] and the factors of [latex]2[\/latex] are [latex]\\pm1[\/latex] and [latex]\\pm2[\/latex]. The possible values for [latex]\\frac{p}{q}[\/latex] are [latex]\\pm1[\/latex] and [latex]\\pm\\frac{1}{2}[\/latex]. These are the possible rational zeros for the function. We can determine which of the possible zeros are actual zeros by substituting these values for [latex]x[\/latex] in [latex]f(x)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f(-1) = 2(-1)^3 + (-1)^2 - 4(-1) + 1 = 4[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]f(1) = 2(1)^3 + (1)^2 - 4(1) + 1 = 0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(-\\frac{1}{2}\\right) = 2\\left(-\\frac{1}{2}\\right)^3 + \\left(-\\frac{1}{2}\\right)^2 - 4\\left(-\\frac{1}{2}\\right) + 1 = 3[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(\\frac{1}{2}\\right) = 2\\left(\\frac{1}{2}\\right)^3 + \\left(\\frac{1}{2}\\right)^2 - 4\\left(\\frac{1}{2}\\right) + 1 = -\\frac{1}{2}[\/latex]<\/p>\n<p>Of those, [latex]-1, -\\frac{1}{2},[\/latex] and [latex]\\frac{1}{2}[\/latex] are not zeros of [latex]f(x)[\/latex]. [latex]1[\/latex] is the only rational zero of [latex]f(x)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Use the Rational Zero Theorem to find the rational zeros of [latex]f(x)=x^3\u22125x^2+2x+1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q981143\">Show Solution<\/button> <\/p>\n<div id=\"q981143\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are no rational zeros.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video see more on the rational zero theorem.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328535&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=Iaq7z7reznM&amp;video_target=tpm-plugin-3vmzk5cw-Iaq7z7reznM\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Finding+All+Zeros+of+a+Polynomial+Function+Using+The+Rational+Zero+Theorem.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding All Zeros of a Polynomial Function Using The Rational Zero Theorem\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Graphs of Polynomial Functions<\/h2>\n<h3>Zeros and Multiplicity<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>The zeros of a polynomial function are the points where the graph crosses the x-axis, meaning the output [latex]y[\/latex] is zero. For example, the polynomial function [latex]f(x)=(x+3)(x\u22122)^2(x+1)^3[\/latex]has zeros at:<\/p>\n<ul>\n<li>[latex]x = -3[\/latex]<\/li>\n<li>[latex]x = 2[\/latex]<\/li>\n<li>[latex]x = -1[\/latex]<\/li>\n<\/ul>\n<p><strong>Quick Tips: Multiplicity and Graph Behavior<\/strong><\/p>\n<ul>\n<li>Single Zeros: When a factor appears once, such as [latex]x+3[\/latex], the graph crosses the [latex]x[\/latex]-axis at this zero. This is a zero with multiplicity [latex]1[\/latex].<\/li>\n<li>Even Multiplicity: Zeros with an even multiplicity, like [latex]x=2[\/latex] in [latex](x\u22122)^2[\/latex], indicate the graph touches the x-axis and turns around.<\/li>\n<li>Odd Multiplicity: Zeros with an odd multiplicity, such as [latex]x=\u22121[\/latex] in [latex](x+1)^3[\/latex], show that the graph will cross the [latex]x[\/latex]-axis.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Use the graph of the function of degree [latex]5[\/latex] to identify the zeros of the function and their multiplicities.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"small aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201600\/CNX_Precalc_Figure_03_04_0102.jpg\" alt=\"Graph of an even-degree polynomial with degree 6.\" width=\"487\" height=\"253\" \/><\/div>\n<p>&nbsp;<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q874458\">Show Solution<\/button><\/p>\n<div id=\"q874458\" class=\"hidden-answer\" style=\"display: none\">\n<p>The graph has a zero of [latex]\u20135[\/latex] with multiplicity [latex]1[\/latex], a zero of [latex]\u20131[\/latex] with multiplicity [latex]2[\/latex], and a zero of [latex]3[\/latex] with multiplicity [latex]2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Use the graph below to find the zeros of the degree [latex]6[\/latex] function and their multiplicities.<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-6767\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/11\/08231949\/Screen-Shot-2019-07-08-at-4.19.09-PM.png\" alt=\"Graph of a polynomial that goes through\/touches the x-axis at (-3,0), (-2,0), and (1,0)\" width=\"503\" height=\"494\" \/><\/div>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q953557\">Show Solution<\/button><\/p>\n<div id=\"q953557\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<td>Zero<\/td>\n<td>Multiplicity<\/td>\n<\/tr>\n<tr>\n<td>[latex](-3,0)[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex](-2,0)[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex](1,0)[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video see an example of finding the zeros and their multiplicities.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328536&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=VP_7Zy1YDE4&amp;video_target=tpm-plugin-k2nntq0b-VP_7Zy1YDE4\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/How+to+Determine+the+Multiplicity+and+Zeros+of+a+Polynomial.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Determine the Multiplicity and Zeros of a Polynomial\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Determining the Number of Turning Points and Intercepts from the Degree of the Polynomial<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>The number of turning points in a polynomial graph is related to the degree of the polynomial. A polynomial of degree [latex]n[\/latex] can have up to [latex]n\u22121[\/latex] turning points. For instance, a [latex]4[\/latex]th-degree polynomial can have up to [latex]3[\/latex] turning points. This is an essential concept in visualizing the shape of the graph.<\/p>\n<p>The degree and leading coefficient of a polynomial function influence its end behavior. For a polynomial [latex]f(x)=ax^n[\/latex]:<\/p>\n<ul>\n<li>If [latex]n[\/latex] is odd and [latex]a>0[\/latex], the ends of the graph go in opposite directions, with the right end going up.<\/li>\n<li>If [latex]n[\/latex] is even and [latex]a>0[\/latex], both ends of the graph go up.<\/li>\n<\/ul>\n<p>This knowledge helps in sketching a rough graph of the polynomial function.<\/p><\/div>\n<section class=\"textbox example\">\n<p>Without graphing the function, determine the maximum number of <em>x<\/em>-intercepts and turning points for [latex]f\\left(x\\right)=108 - 13{x}^{9}-8{x}^{4}+14{x}^{12}+2{x}^{3}[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q304362\">Show Solution<\/button><\/p>\n<div id=\"q304362\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are at most [latex]12[\/latex] [latex]x[\/latex]-intercepts and at most [latex]11[\/latex] turning points.<\/p><\/div>\n<\/div>\n<\/section>\n<p>The following video gives a 5 minute lesson on how to determine the number of intercepts and turning points of a polynomial function given its degree.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Turning Points and X Intercepts of a Polynomial Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/9WW0EetLD4Q?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Turning+Points+and+X+Intercepts+of+a+Polynomial+Function.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cTurning Points and X Intercepts of a Polynomial Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>&nbsp;<\/h2>\n<h3>Identifying&nbsp;the Shape of the Graph of a Polynomial Function<\/h3>\n<p>Polynomial functions are like the diverse landscapes of mathematics, where each function can represent a unique terrain. The graphs of these functions are smooth and continuous, meaning they have no sharp turns or breaks. Just like a rolling hillside, these functions have a fluidity that&#8217;s both predictable and elegant.<\/p>\n<section class=\"textbox example\">\n<p>Which of the graphs below&nbsp;represents a polynomial function?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201420\/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><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q207827\">Show Solution<\/button><\/p>\n<div id=\"q207827\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134129608\">The graphs of [latex]f[\/latex]&nbsp;and [latex]h[\/latex] are graphs of polynomial functions. They are smooth and <strong>continuous<\/strong>.<\/p>\n<p id=\"fs-id1165134188794\">The graphs of [latex]g[\/latex]&nbsp;and [latex]k[\/latex] are graphs of functions that are not polynomials. The graph of function [latex]g[\/latex]&nbsp;has a sharp corner. The graph of function [latex]k[\/latex]&nbsp;is not continuous.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h4>Identifying End Behavior of Polynomial Functions<\/h4>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p><strong>Quick Tips: Even vs. Odd Degree Polynomials<\/strong><\/p>\n<ul>\n<li><strong>Even Degree Polynomials<\/strong>: Think of even-degree polynomials like [latex]f(x)=x^2, g(x)=x^4,[\/latex] and [latex]h(x)=x^6[\/latex] as the serene hills that either rise or fall on both ends. As the degree increases, the middle part of the graph becomes flatter, and the ends become steeper, much like a valley becoming more pronounced with time.<\/li>\n<li><strong>Odd Degree Polynomials<\/strong>: Odd-degree polynomials such as [latex]f(x)=x^3, g(x)=x^5,[\/latex] and [latex]h(x)=x^7[\/latex] are the adventurous mountain ranges of the polynomial world. One side ascends to the sky, while the other descends into the depths of the ocean. This is due to the fact that a negative input will yield a negative output when the degree is odd, creating a graph that has a perfect balance of ascent and descent.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Describe the end behavior of the polynomial function in the graph below.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194522\/CNX_Precalc_Figure_03_03_016n2.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"440\" \/><\/div>\n<p>&nbsp;<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q304329\">Show Solution<\/button><\/p>\n<div id=\"q304329\" class=\"hidden-answer\" style=\"display: none\">\n<p>As [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty[\/latex] and as [latex]x\\to -\\infty , f\\left(x\\right)\\to -\\infty[\/latex]. It has the shape of an even degree power function with a negative coefficient.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Given the function [latex]f\\left(x\\right)=0.2\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 5\\right)[\/latex], express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q657153\">Show Solution<\/button><\/p>\n<div id=\"q657153\" class=\"hidden-answer\" style=\"display: none\">\n<p>The leading term is [latex]0.2{x}^{3}[\/latex], so it is a degree [latex]3[\/latex] polynomial. As [latex]x[\/latex] approaches positive infinity, [latex]f\\left(x\\right)[\/latex] increases without bound; as [latex]x[\/latex] approaches negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound.<\/p><\/div>\n<\/div>\n<\/section>\n<p>In the following video, you&#8217;ll see more examples that summarize the end behavior of polynomial functions and which components of the function contribute to it.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Summary of End Behavior or Long Run Behavior of Polynomial Functions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/y78Dpr9LLN0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Summary+of+End+Behavior+or+Long+Run+Behavior+of+Polynomial+Functions.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSummary of End Behavior or Long Run Behavior of Polynomial Functions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h4>The Sign of the Leading Term<\/h4>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>The leading term of a polynomial function is like the compass that dictates the direction of your graph. Changing the sign of this term is akin to entering a mirror world, where everything is flipped. <strong>Quick Tips: Even vs. Odd Degree Polynomials<\/strong><\/p>\n<ul>\n<li>Positive Leading Term: When the leading term is positive, as in 3x^4[\/latex], the arms of the graph reach for the heavens. The graph stays positive, no matter if [latex]x[\/latex] is negative or positive.<\/li>\n<li>Negative Leading Term: Introduce a negative sign, and the graph of [latex]\u22123x^4[\/latex] turns upside down, diving down as if reflecting in a lake. The negative leading term ensures that the arms of the polynomial are pointing downwards, regardless of the value of [latex]x[\/latex].<\/li>\n<\/ul>\n<\/div>\n<p>The table below summarizes all four cases mentioned.<\/p>\n<table>\n<thead>\n<tr>\n<th style=\"text-align: center;\">Even Degree<\/th>\n<th style=\"text-align: center;\">Odd Degree<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/11.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12504\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201602\/11.png\" alt=\"11\" width=\"423\" height=\"559\" \/><\/a><\/td>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/12.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12505\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201605\/12.png\" alt=\"12\" width=\"397\" height=\"560\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/13.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12506\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201607\/13.png\" alt=\"13\" width=\"387\" height=\"574\" \/><\/a><\/td>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/14.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12507\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201609\/14.png\" alt=\"14\" width=\"404\" height=\"564\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Sketching Graphs<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/p>\n<p><\/strong>When we sketch the graphs of polynomial functions, we&#8217;re essentially drawing a rough map of how the function behaves. It&#8217;s like sketching the path of a hiker in the mountains: we mark the starting point, the peaks, and the valleys. Here&#8217;s how we do it:<\/p>\n<ul>\n<li><strong>Intercepts<\/strong>: These are our starting points. Just like a hiker checks a map for where to begin, we find where our graph crosses the axes.<\/li>\n<li><strong>Symmetry<\/strong>: Is our path mirrored on either side? If our function is even, the graph is symmetric like butterfly wings across the [latex]y[\/latex]-axis ([latex]f(\u2212x)=f(x)[\/latex]). If it&#8217;s odd, it&#8217;s like a twisty mountain road that flips across the origin ([latex]f(\u2212x)=\u2212f(x)[\/latex]).<\/li>\n<li><strong>Multiplicities and Intercepts<\/strong>: This tells us how our hiker approaches each peak or valley. Does she pass straight through, or does she hesitate and turn back a bit? The multiplicity of zeros gives us this behavior at [latex]x[\/latex]-intercepts.<\/li>\n<li><strong>End Behavior<\/strong>: This is the direction our hiker heads off into the distance. It&#8217;s determined by the leading term of our polynomial.<\/li>\n<li><strong>Sketching the Graph<\/strong>: With all this information, we can draw a rough trail of our function, ensuring we don&#8217;t have more twists and turns (turning points) than the degree of the polynomial allows.<\/li>\n<li><strong>Technology Check<\/strong>: Just like a hiker might check GPS, we can use graphing technology to confirm our path.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">Sketch a possible graph for [latex]f\\left(x\\right)=\\frac{1}{4}x{\\left(x - 1\\right)}^{4}{\\left(x+3\\right)}^{3}[\/latex]. Check yourself with an online graphing calculator when you are done. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q812296\">Show Solution<\/button> <\/p>\n<div id=\"q812296\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/30233150\/CNX_Precalc_Figure_03_04_0212.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2911\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/30233150\/CNX_Precalc_Figure_03_04_0212.jpg\" alt=\"Graph of f(x)=(1\/4)x(x-1)^4(x+3)^3.\" width=\"487\" height=\"366\" \/><\/a><\/div>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video to see an example of sketching a polynomial.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328600&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=PWLvZZfpC5I&amp;video_target=tpm-plugin-g3fbwvu9-PWLvZZfpC5I\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/How+to+Sketch+a+Polynomial+Function.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Sketch a Polynomial Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>The following video is a bit long, but it covers how to graph a polynomial function using end behavior, multiplicity, and zeros.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328601&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=a5x4lwnvHM0&amp;video_target=tpm-plugin-2oau5nr7-a5x4lwnvHM0\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/How+To+Graph+Polynomial+Functions+Using+End+Behavior%2C+Multiplicity+%26+Zeros.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow To Graph Polynomial Functions Using End Behavior, Multiplicity &amp; Zeros\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Writing Formulas for Polynomial Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/p>\n<p><\/strong>Writing the formula for a polynomial function from its graph is like retracing the hiker&#8217;s steps to map her exact route:<\/p>\n<ul>\n<li><strong>[latex]X[\/latex]-Intercepts<\/strong>: Find where the hiker started and finished her journey. These give us the factors of the polynomial.<\/li>\n<li><strong>Multiplicity<\/strong>: How did she approach each peak? Did she go straight over, or did she pause and reflect? This tells us the multiplicity of each factor.<\/li>\n<li><strong>Least Degree<\/strong>: We want the simplest map that still shows all the important points. Find the polynomial of the least degree that includes all factors.<\/li>\n<li><strong>Stretch Factor<\/strong>: How steep were the mountains? Use another point (like the [latex]y[\/latex]-intercept) to figure out the &#8220;stretch&#8221; of our polynomial, which scales the function up or down.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Given the graph below, write a formula for the function shown.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201629\/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\" \/><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q427364\">Show Solution<\/button> <\/p>\n<div id=\"q427364\" 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<\/section>\n<p>Watch the following video to see an example of writing the equation of a polynomial.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/m9PM6Z2b55A?si=4QJXnusXTgfdJ86w\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex1_+Find+an+Equation+of+a+Degree+4+Polynomial+Function+From+the+Graph+of+the+Function.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx1: Find an Equation of a Degree 4 Polynomial Function From the Graph of the Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":6,"menu_order":47,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":185,"module-header":"","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/238"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/238\/revisions"}],"predecessor-version":[{"id":307,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/238\/revisions\/307"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/parts\/185"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/238\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/media?parent=238"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapter-type?post=238"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/contributor?post=238"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/license?post=238"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}