{"id":1881,"date":"2024-06-18T17:47:18","date_gmt":"2024-06-18T17:47:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1881"},"modified":"2025-01-15T15:11:17","modified_gmt":"2025-01-15T15:11:17","slug":"module-9-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/module-9-cheat-sheet\/","title":{"raw":"Power and Polynomial Functions: Cheat Sheet","rendered":"Power and Polynomial Functions: Cheat Sheet"},"content":{"raw":"<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+LOHM\/Cheat+Sheets\/College+Algebra+Cheat+Sheet+M9_+Power+and+Polynomial+Functions.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\r\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\r\n\r\n<h2>Essential Concepts<\/h2>\r\n<h3><span data-sheets-root=\"1\">Introduction to Power and Polynomial Functions<\/span><\/h3>\r\n<ul id=\"fs-id1165135438864\">\r\n \t<li>A power function is a variable base raised to a number power.<\/li>\r\n \t<li>The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.<\/li>\r\n \t<li>The end behavior depends on whether the power is even or odd.<\/li>\r\n \t<li>A polynomial function is the sum of terms, each of which consists of a transformed power function with non-negative integer powers.<\/li>\r\n \t<li>The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.<\/li>\r\n \t<li>The sign of the leading term will determine the direction of the ends of the graph:\r\n<ul>\r\n \t<li>even degree and positive coefficient: both ends point up<\/li>\r\n \t<li>even degree and negative coefficient: both ends point down<\/li>\r\n \t<li>odd degree and positive coefficient: the left-most end points down and the right-most end points up.<\/li>\r\n \t<li>odd degree and negative coefficient: the left-most end points up and the right-most end points down.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function.<\/li>\r\n \t<li>A polynomial of degree [latex]n[\/latex]\u00a0will have at most [latex]n[\/latex]\u00a0[latex]x[\/latex]-intercepts and at most [latex]n \u2013 1[\/latex]\u00a0turning points.<\/li>\r\n<\/ul>\r\n<h3><span data-sheets-root=\"1\">Graphs of Polynomial Functions<\/span><\/h3>\r\n<ul>\r\n \t<li>Polynomial functions of degree 2 or more are smooth, continuous functions.<\/li>\r\n \t<li>To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.<\/li>\r\n \t<li>Another way to find the [latex]x[\/latex]<em>-<\/em>intercepts of a polynomial function is to graph the function and identify the points where the graph crosses the [latex]x[\/latex]-axis.<\/li>\r\n \t<li>The multiplicity of a zero determines how the graph behaves at the\u00a0[latex]x[\/latex]-intercept.\r\n<ul id=\"fs-id1165135438864\">\r\n \t<li>The graph of a polynomial will cross the [latex]x[\/latex]-axis at a zero with odd multiplicity.<\/li>\r\n \t<li>The graph of a polynomial will touch and bounce off the [latex]x[\/latex]-axis at a zero with even multiplicity.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The graph of a polynomial will cross the [latex]x[\/latex]-axis at a zero with odd multiplicity.<\/li>\r\n \t<li>The graph of a polynomial will touch and bounce off the [latex]x[\/latex]-axis at a zero with even multiplicity.<\/li>\r\n \t<li>The end behavior of a polynomial function depends on the leading term.<\/li>\r\n \t<li>The graph of a polynomial function changes direction at its turning points.<\/li>\r\n \t<li>To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most [latex]n \u2013\u00a01[\/latex] turning points.<\/li>\r\n \t<li>Graphing a polynomial function helps to estimate local and global extremas.<\/li>\r\n \t<li>The Intermediate Value Theorem tells us that if [latex]f\\left(a\\right) \\text{and} f\\left(b\\right)[\/latex]\u00a0have opposite signs, then there exists at least one value [latex]c[\/latex]\u00a0between [latex]a[\/latex]\u00a0and [latex]b[\/latex]\u00a0for which [latex]f\\left(c\\right)=0[\/latex].<\/li>\r\n<\/ul>\r\n<h3><span data-sheets-root=\"1\">Dividing Polynomials<\/span><\/h3>\r\n<ul>\r\n \t<li>Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree.<\/li>\r\n \t<li>The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.<\/li>\r\n \t<li>Synthetic division is a shortcut that can be used to divide a polynomial by a binomial of the form [latex]x \u2013\u00a0k[\/latex].<\/li>\r\n \t<li>Polynomial division can be used to solve application problems, including area and volume.<\/li>\r\n<\/ul>\r\n<h3><span data-sheets-root=\"1\">Zeros of Polynomial Functions<\/span><\/h3>\r\n<ul id=\"fs-id1165135380122\">\r\n \t<li>To find [latex]f\\left(k\\right)[\/latex], determine the remainder of the polynomial [latex]f\\left(x\\right)[\/latex] when it is divided by [latex]x-k[\/latex].<\/li>\r\n \t<li>[latex]k[\/latex]\u00a0is a zero of [latex]f\\left(x\\right)[\/latex] if and only if [latex]\\left(x-k\\right)[\/latex] \u00a0is a factor of [latex]f\\left(x\\right)[\/latex].<\/li>\r\n \t<li>Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient.<\/li>\r\n \t<li>When the leading coefficient is [latex]1[\/latex], the possible rational zeros are the factors of the constant term.<\/li>\r\n \t<li>Synthetic division can be used to find the zeros of a polynomial function.<\/li>\r\n \t<li>According to the Fundamental Theorem of Algebra, every polynomial function has at least one complex zero.<\/li>\r\n \t<li>Every polynomial function with degree greater than [latex]0[\/latex] has at least one complex zero.<\/li>\r\n \t<li>Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form [latex]\\left(x-c\\right)[\/latex] where <em>c<\/em>\u00a0is a complex number.<\/li>\r\n \t<li>The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer.<\/li>\r\n \t<li>The number of negative real zeros of a polynomial function is either the number of sign changes of [latex]f\\left(-x\\right)[\/latex] \u00a0or less than the number of sign changes by an even integer.<\/li>\r\n \t<li>Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division.<\/li>\r\n \t<li>Remainder Theorem states that if a polynomial [latex]f\\left(x\\right)[\/latex] is divided by [latex]x-k[\/latex] , then the remainder is equal to the value [latex]f\\left(k\\right)[\/latex].<\/li>\r\n \t<li>Rational Zero Theorem states that the possible rational zeros of a polynomial function have the form [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>\r\n \t<li>Factor Theorem states that [latex]k[\/latex] is a zero of polynomial function [latex]f\\left(x\\right)[\/latex] if and only if [latex]\\left(x-k\\right)[\/latex] \u00a0is a factor of [latex]f\\left(x\\right)[\/latex].<\/li>\r\n \t<li>The Fundamental Theorem of Algebra states that a polynomial function with degree greater than [latex]0[\/latex] has at least one complex zero.<\/li>\r\n \t<li>Linear Factorization Theorem states that a polynomial function will have the same number of factors as its degree, and each factor will be in the form [latex]\\left(x-c\\right)[\/latex] where c is a complex number.<\/li>\r\n \t<li>Complex Conjugates Theorem states that if the polynomial function [latex]f[\/latex] has real coefficients and a complex zero of the form [latex]a+bi[\/latex], then the complex conjugate of the zero, [latex]a\u2212bi[\/latex], is also a zero.<\/li>\r\n \t<li>Descartes\u2019 Rule of Signs is a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of [latex]f\\left(x\\right)[\/latex] and [latex]f\\left(-x\\right)[\/latex].<\/li>\r\n<\/ul>\r\n<ul id=\"fs-id1165135438864\"><\/ul>\r\n<h2>Key Equations<\/h2>\r\n<table id=\"eip-id1165134063974\" style=\"width: 89.2119%;\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 25.8888%;\">general form of a polynomial function<\/td>\r\n<td style=\"width: 102.37%;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25.8888%;\">Division Algorithm<\/td>\r\n<td style=\"width: 102.37%;\">[latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex] where [latex]q\\left(x\\right)\\ne 0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165137668266\" class=\"definition\">\r\n \t<dt><strong>coefficient<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135194915\">a nonzero real number multiplied by a variable raised to an exponent<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135194918\" class=\"definition\">\r\n \t<dt><strong>continuous function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135194921\">a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137832108\" class=\"definition\">\r\n \t<dt><strong>degree<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137832112\">the highest power of the variable that occurs in a polynomial<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133281424\" class=\"definition\">\r\n \t<dt><strong>Descartes\u2019 Rule of Signs<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165133281430\">a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of [latex]f\\left(x\\right)[\/latex] and [latex]f\\left(-x\\right)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943522\" class=\"definition\">\r\n \t<dt><strong>Division Algorithm<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">given a polynomial dividend [latex]f\\left(x\\right)[\/latex]\u00a0and a non-zero polynomial divisor [latex]d\\left(x\\right)[\/latex]\u00a0where the degree of [latex]d\\left(x\\right)[\/latex]\u00a0is less than or equal to the degree of [latex]f\\left(x\\right)[\/latex],\u00a0there exist unique polynomials [latex]q\\left(x\\right)[\/latex]\u00a0and [latex]r\\left(x\\right)[\/latex]\u00a0such that [latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex]\u00a0where [latex]q\\left(x\\right)[\/latex]\u00a0is the quotient and [latex]r\\left(x\\right)[\/latex]\u00a0is the remainder. The remainder is either equal to zero or has degree strictly less than [latex]d\\left(x\\right)[\/latex].<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137832115\" class=\"definition\">\r\n \t<dt><strong>end behavior<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165131990654\">the behavior of the graph of a function as the input decreases without bound and increases without bound<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135459801\" class=\"definition\">\r\n \t<dt><strong>Factor Theorem<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135459806\"><em>k<\/em>\u00a0is a zero of polynomial function [latex]f\\left(x\\right)[\/latex] if and only if [latex]\\left(x-k\\right)[\/latex] \u00a0is a factor of [latex]f\\left(x\\right)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133045332\" class=\"definition\">\r\n \t<dt><strong>Fundamental Theorem of Algebra<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165133045337\">a polynomial function with degree greater than 0 has at least one complex zero<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943522\" class=\"definition\">\r\n \t<dt><strong>global maximum<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">highest turning point on a graph; [latex]f\\left(a\\right)[\/latex]\u00a0where [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex]\u00a0for all [latex]x[\/latex].<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943522\" class=\"definition\">\r\n \t<dt><b>global minimum<\/b><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">lowest turning point on a graph; [latex]f\\left(a\\right)[\/latex]\u00a0where [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex] for all [latex]x[\/latex].<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943522\" class=\"definition\">\r\n \t<dt><strong>Intermediate Value Theorem<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">for two numbers [latex]a[\/latex]\u00a0and [latex]b[\/latex]\u00a0in the domain of [latex]f[\/latex],\u00a0if [latex]a&lt;b[\/latex]\u00a0and [latex]f\\left(a\\right)\\ne f\\left(b\\right)[\/latex],\u00a0then the function [latex]f[\/latex]\u00a0takes on every value between [latex]f\\left(a\\right)[\/latex]\u00a0and [latex]f\\left(b\\right)[\/latex];\u00a0specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the [latex]x[\/latex]-axis<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131990658\" class=\"definition\">\r\n \t<dt><strong>leading coefficient<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165131990661\">the coefficient of the leading term<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943522\" class=\"definition\">\r\n \t<dt><strong>leading term<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">the term containing the highest power of the variable<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133045341\" class=\"definition\">\r\n \t<dt><strong>Linear Factorization Theorem<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165133045347\">allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form [latex]\\left(x-c\\right)[\/latex] where <em>c<\/em>\u00a0is a complex number<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943522\" class=\"definition\">\r\n \t<dt><strong>multiplicity<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], [latex]x=h[\/latex]\u00a0is a zero of multiplicity [latex]p[\/latex].<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943528\" class=\"definition\">\r\n \t<dt><strong>polynomial function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134297639\">a function that consists of either zero or the sum of a finite number of non-zero\u00a0terms, each of which is a product of a number, called the\u00a0coefficient\u00a0of the term, and a variable raised to a non-negative integer power.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134297646\" class=\"definition\">\r\n \t<dt><strong>power function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135486042\">a function that can be represented in the form [latex]f\\left(x\\right)=a{x}^{n}[\/latex]\u00a0where <em>a\u00a0<\/em>is a constant, the base is a variable, and the exponent is\u00a0<i>n<\/i>,\u00a0is a smooth curve represented by a graph with no sharp corners<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135456904\" class=\"definition\">\r\n \t<dt><strong>Rational Zero Theorem<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135456910\">the possible rational zeros of a polynomial function have the form [latex]\\frac{p}{q}[\/latex] where <em>p<\/em>\u00a0is a factor of the constant term and <em>q<\/em>\u00a0is a factor of the leading coefficient<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137938597\" class=\"definition\">\r\n \t<dt><strong>Remainder Theorem<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137938602\">if a polynomial [latex]f\\left(x\\right)[\/latex] is divided by [latex]x-k[\/latex] , then the remainder is equal to the value [latex]f\\left(k\\right)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943522\" class=\"definition\">\r\n \t<dt><strong>synthetic division<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">a shortcut method that can be used to divide a polynomial by a binomial of the form [latex]x \u2013 k[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137644987\" class=\"definition\">\r\n \t<dt><strong>term of a polynomial function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137644990\">any [latex]{a}_{i}{x}^{i}[\/latex]\u00a0of a polynomial function in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133085661\" class=\"definition\">\r\n \t<dt><strong>turning point<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165133085665\">the location where the graph of a function changes direction<\/dd>\r\n<\/dl>","rendered":"<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+LOHM\/Cheat+Sheets\/College+Algebra+Cheat+Sheet+M9_+Power+and+Polynomial+Functions.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\n<h2>Essential Concepts<\/h2>\n<h3><span data-sheets-root=\"1\">Introduction to Power and Polynomial Functions<\/span><\/h3>\n<ul id=\"fs-id1165135438864\">\n<li>A power function is a variable base raised to a number power.<\/li>\n<li>The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.<\/li>\n<li>The end behavior depends on whether the power is even or odd.<\/li>\n<li>A polynomial function is the sum of terms, each of which consists of a transformed power function with non-negative integer powers.<\/li>\n<li>The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.<\/li>\n<li>The sign of the leading term will determine the direction of the ends of the graph:\n<ul>\n<li>even degree and positive coefficient: both ends point up<\/li>\n<li>even degree and negative coefficient: both ends point down<\/li>\n<li>odd degree and positive coefficient: the left-most end points down and the right-most end points up.<\/li>\n<li>odd degree and negative coefficient: the left-most end points up and the right-most end points down.<\/li>\n<\/ul>\n<\/li>\n<li>The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function.<\/li>\n<li>A polynomial of degree [latex]n[\/latex]\u00a0will have at most [latex]n[\/latex]\u00a0[latex]x[\/latex]-intercepts and at most [latex]n \u2013 1[\/latex]\u00a0turning points.<\/li>\n<\/ul>\n<h3><span data-sheets-root=\"1\">Graphs of Polynomial Functions<\/span><\/h3>\n<ul>\n<li>Polynomial functions of degree 2 or more are smooth, continuous functions.<\/li>\n<li>To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.<\/li>\n<li>Another way to find the [latex]x[\/latex]<em>&#8211;<\/em>intercepts of a polynomial function is to graph the function and identify the points where the graph crosses the [latex]x[\/latex]-axis.<\/li>\n<li>The multiplicity of a zero determines how the graph behaves at the\u00a0[latex]x[\/latex]-intercept.\n<ul>\n<li>The graph of a polynomial will cross the [latex]x[\/latex]-axis at a zero with odd multiplicity.<\/li>\n<li>The graph of a polynomial will touch and bounce off the [latex]x[\/latex]-axis at a zero with even multiplicity.<\/li>\n<\/ul>\n<\/li>\n<li>The graph of a polynomial will cross the [latex]x[\/latex]-axis at a zero with odd multiplicity.<\/li>\n<li>The graph of a polynomial will touch and bounce off the [latex]x[\/latex]-axis at a zero with even multiplicity.<\/li>\n<li>The end behavior of a polynomial function depends on the leading term.<\/li>\n<li>The graph of a polynomial function changes direction at its turning points.<\/li>\n<li>To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most [latex]n \u2013\u00a01[\/latex] turning points.<\/li>\n<li>Graphing a polynomial function helps to estimate local and global extremas.<\/li>\n<li>The Intermediate Value Theorem tells us that if [latex]f\\left(a\\right) \\text{and} f\\left(b\\right)[\/latex]\u00a0have opposite signs, then there exists at least one value [latex]c[\/latex]\u00a0between [latex]a[\/latex]\u00a0and [latex]b[\/latex]\u00a0for which [latex]f\\left(c\\right)=0[\/latex].<\/li>\n<\/ul>\n<h3><span data-sheets-root=\"1\">Dividing Polynomials<\/span><\/h3>\n<ul>\n<li>Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree.<\/li>\n<li>The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.<\/li>\n<li>Synthetic division is a shortcut that can be used to divide a polynomial by a binomial of the form [latex]x \u2013\u00a0k[\/latex].<\/li>\n<li>Polynomial division can be used to solve application problems, including area and volume.<\/li>\n<\/ul>\n<h3><span data-sheets-root=\"1\">Zeros of Polynomial Functions<\/span><\/h3>\n<ul id=\"fs-id1165135380122\">\n<li>To find [latex]f\\left(k\\right)[\/latex], determine the remainder of the polynomial [latex]f\\left(x\\right)[\/latex] when it is divided by [latex]x-k[\/latex].<\/li>\n<li>[latex]k[\/latex]\u00a0is a zero of [latex]f\\left(x\\right)[\/latex] if and only if [latex]\\left(x-k\\right)[\/latex] \u00a0is a factor of [latex]f\\left(x\\right)[\/latex].<\/li>\n<li>Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient.<\/li>\n<li>When the leading coefficient is [latex]1[\/latex], the possible rational zeros are the factors of the constant term.<\/li>\n<li>Synthetic division can be used to find the zeros of a polynomial function.<\/li>\n<li>According to the Fundamental Theorem of Algebra, every polynomial function has at least one complex zero.<\/li>\n<li>Every polynomial function with degree greater than [latex]0[\/latex] has at least one complex zero.<\/li>\n<li>Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form [latex]\\left(x-c\\right)[\/latex] where <em>c<\/em>\u00a0is a complex number.<\/li>\n<li>The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer.<\/li>\n<li>The number of negative real zeros of a polynomial function is either the number of sign changes of [latex]f\\left(-x\\right)[\/latex] \u00a0or less than the number of sign changes by an even integer.<\/li>\n<li>Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division.<\/li>\n<li>Remainder Theorem states that if a polynomial [latex]f\\left(x\\right)[\/latex] is divided by [latex]x-k[\/latex] , then the remainder is equal to the value [latex]f\\left(k\\right)[\/latex].<\/li>\n<li>Rational Zero Theorem states that the possible rational zeros of a polynomial function have the form [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>Factor Theorem states that [latex]k[\/latex] is a zero of polynomial function [latex]f\\left(x\\right)[\/latex] if and only if [latex]\\left(x-k\\right)[\/latex] \u00a0is a factor of [latex]f\\left(x\\right)[\/latex].<\/li>\n<li>The Fundamental Theorem of Algebra states that a polynomial function with degree greater than [latex]0[\/latex] has at least one complex zero.<\/li>\n<li>Linear Factorization Theorem states that a polynomial function will have the same number of factors as its degree, and each factor will be in the form [latex]\\left(x-c\\right)[\/latex] where c is a complex number.<\/li>\n<li>Complex Conjugates Theorem states that if the polynomial function [latex]f[\/latex] has real coefficients and a complex zero of the form [latex]a+bi[\/latex], then the complex conjugate of the zero, [latex]a\u2212bi[\/latex], is also a zero.<\/li>\n<li>Descartes\u2019 Rule of Signs is a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of [latex]f\\left(x\\right)[\/latex] and [latex]f\\left(-x\\right)[\/latex].<\/li>\n<\/ul>\n<ul><\/ul>\n<h2>Key Equations<\/h2>\n<table id=\"eip-id1165134063974\" style=\"width: 89.2119%;\" summary=\"..\">\n<tbody>\n<tr>\n<td style=\"width: 25.8888%;\">general form of a polynomial function<\/td>\n<td style=\"width: 102.37%;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25.8888%;\">Division Algorithm<\/td>\n<td style=\"width: 102.37%;\">[latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex] where [latex]q\\left(x\\right)\\ne 0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137668266\" class=\"definition\">\n<dt><strong>coefficient<\/strong><\/dt>\n<dd id=\"fs-id1165135194915\">a nonzero real number multiplied by a variable raised to an exponent<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135194918\" class=\"definition\">\n<dt><strong>continuous function<\/strong><\/dt>\n<dd id=\"fs-id1165135194921\">a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137832108\" class=\"definition\">\n<dt><strong>degree<\/strong><\/dt>\n<dd id=\"fs-id1165137832112\">the highest power of the variable that occurs in a polynomial<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133281424\" class=\"definition\">\n<dt><strong>Descartes\u2019 Rule of Signs<\/strong><\/dt>\n<dd id=\"fs-id1165133281430\">a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of [latex]f\\left(x\\right)[\/latex] and [latex]f\\left(-x\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n<dt><strong>Division Algorithm<\/strong><\/dt>\n<dd id=\"fs-id1165132943525\">given a polynomial dividend [latex]f\\left(x\\right)[\/latex]\u00a0and a non-zero polynomial divisor [latex]d\\left(x\\right)[\/latex]\u00a0where the degree of [latex]d\\left(x\\right)[\/latex]\u00a0is less than or equal to the degree of [latex]f\\left(x\\right)[\/latex],\u00a0there exist unique polynomials [latex]q\\left(x\\right)[\/latex]\u00a0and [latex]r\\left(x\\right)[\/latex]\u00a0such that [latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex]\u00a0where [latex]q\\left(x\\right)[\/latex]\u00a0is the quotient and [latex]r\\left(x\\right)[\/latex]\u00a0is the remainder. The remainder is either equal to zero or has degree strictly less than [latex]d\\left(x\\right)[\/latex].<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137832115\" class=\"definition\">\n<dt><strong>end behavior<\/strong><\/dt>\n<dd id=\"fs-id1165131990654\">the behavior of the graph of a function as the input decreases without bound and increases without bound<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135459801\" class=\"definition\">\n<dt><strong>Factor Theorem<\/strong><\/dt>\n<dd id=\"fs-id1165135459806\"><em>k<\/em>\u00a0is a zero of polynomial function [latex]f\\left(x\\right)[\/latex] if and only if [latex]\\left(x-k\\right)[\/latex] \u00a0is a factor of [latex]f\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133045332\" class=\"definition\">\n<dt><strong>Fundamental Theorem of Algebra<\/strong><\/dt>\n<dd id=\"fs-id1165133045337\">a polynomial function with degree greater than 0 has at least one complex zero<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt><strong>global maximum<\/strong><\/dt>\n<dd>highest turning point on a graph; [latex]f\\left(a\\right)[\/latex]\u00a0where [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex]\u00a0for all [latex]x[\/latex].<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt><b>global minimum<\/b><\/dt>\n<dd>lowest turning point on a graph; [latex]f\\left(a\\right)[\/latex]\u00a0where [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex] for all [latex]x[\/latex].<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt><strong>Intermediate Value Theorem<\/strong><\/dt>\n<dd>for two numbers [latex]a[\/latex]\u00a0and [latex]b[\/latex]\u00a0in the domain of [latex]f[\/latex],\u00a0if [latex]a<b[\/latex]\u00a0and [latex]f\\left(a\\right)\\ne f\\left(b\\right)[\/latex],\u00a0then the function [latex]f[\/latex]\u00a0takes on every value between [latex]f\\left(a\\right)[\/latex]\u00a0and [latex]f\\left(b\\right)[\/latex];\u00a0specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the [latex]x[\/latex]-axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131990658\" class=\"definition\">\n<dt><strong>leading coefficient<\/strong><\/dt>\n<dd id=\"fs-id1165131990661\">the coefficient of the leading term<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt><strong>leading term<\/strong><\/dt>\n<dd>the term containing the highest power of the variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133045341\" class=\"definition\">\n<dt><strong>Linear Factorization Theorem<\/strong><\/dt>\n<dd id=\"fs-id1165133045347\">allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form [latex]\\left(x-c\\right)[\/latex] where <em>c<\/em>\u00a0is a complex number<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt><strong>multiplicity<\/strong><\/dt>\n<dd>the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], [latex]x=h[\/latex]\u00a0is a zero of multiplicity [latex]p[\/latex].<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943528\" class=\"definition\">\n<dt><strong>polynomial function<\/strong><\/dt>\n<dd id=\"fs-id1165134297639\">a function that consists of either zero or the sum of a finite number of non-zero\u00a0terms, each of which is a product of a number, called the\u00a0coefficient\u00a0of the term, and a variable raised to a non-negative integer power.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n<dt><strong>power function<\/strong><\/dt>\n<dd id=\"fs-id1165135486042\">a function that can be represented in the form [latex]f\\left(x\\right)=a{x}^{n}[\/latex]\u00a0where <em>a\u00a0<\/em>is a constant, the base is a variable, and the exponent is\u00a0<i>n<\/i>,\u00a0is a smooth curve represented by a graph with no sharp corners<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135456904\" class=\"definition\">\n<dt><strong>Rational Zero Theorem<\/strong><\/dt>\n<dd id=\"fs-id1165135456910\">the possible rational zeros of a polynomial function have the form [latex]\\frac{p}{q}[\/latex] where <em>p<\/em>\u00a0is a factor of the constant term and <em>q<\/em>\u00a0is a factor of the leading coefficient<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137938597\" class=\"definition\">\n<dt><strong>Remainder Theorem<\/strong><\/dt>\n<dd id=\"fs-id1165137938602\">if a polynomial [latex]f\\left(x\\right)[\/latex] is divided by [latex]x-k[\/latex] , then the remainder is equal to the value [latex]f\\left(k\\right)[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt><strong>synthetic division<\/strong><\/dt>\n<dd>a shortcut method that can be used to divide a polynomial by a binomial of the form [latex]x \u2013 k[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n<dt><strong>term of a polynomial function<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">any [latex]{a}_{i}{x}^{i}[\/latex]\u00a0of a polynomial function in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n<dt><strong>turning point<\/strong><\/dt>\n<dd id=\"fs-id1165133085665\">the location where the graph of a function changes direction<\/dd>\n<\/dl>\n","protected":false},"author":12,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":206,"module-header":"cheat_sheet","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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1881"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1881\/revisions"}],"predecessor-version":[{"id":7184,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1881\/revisions\/7184"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/206"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1881\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=1881"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1881"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=1881"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=1881"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}