{"id":16253,"date":"2019-10-02T20:23:01","date_gmt":"2019-10-02T20:23:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-define-and-evaluate-polynomials\/"},"modified":"2024-04-30T21:33:47","modified_gmt":"2024-04-30T21:33:47","slug":"read-define-and-evaluate-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-define-and-evaluate-polynomials\/","title":{"raw":"Defining and Evaluating Polynomials","rendered":"Defining and Evaluating Polynomials"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Define polynomials<\/li>\r\n \t<li>Evaluate a polynomial for a given value<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 id=\"title1\">Identify the terms, the coefficients, and the exponents of a polynomial<\/h2>\r\nPolynomials are algebraic expressions that are created by combining numbers and variables using arithmetic operations such as addition, subtraction, multiplication, division, and exponentiation. A polynomial expression consists of the sum or difference of terms in which each term is a\u00a0real\u00a0number, a variable, or the product of a\u00a0real\u00a0number and variable(s) with a non-negative integer exponents. Non negative integers are\u00a0[latex]0, 1, 2, 3, 4[\/latex], ...\r\n\r\nPolynomials are very useful in applications from science and engineering to business. You may see a resemblance between expressions, which we have been studying in this course, and polynomials. \u00a0Polynomials are a special sub-group of mathematical expressions and equations.\r\n\r\nThe following table is intended to help you tell the difference between what is a polynomial and what is not.\r\n<table>\r\n<thead>\r\n<tr>\r\n<td>IS a Polynomial<\/td>\r\n<td>Is NOT a Polynomial<\/td>\r\n<td>Because<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]2x^2-\\frac{1}{2}x -9[\/latex]<\/td>\r\n<td>[latex]\\frac{2}{x^{2}}+x[\/latex]<\/td>\r\n<td>Polynomials only have variables in the numerator<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\frac{y}{4}-y^3[\/latex]<\/td>\r\n<td>[latex]\\frac{2}{y}+4[\/latex]<\/td>\r\n<td>Polynomials only have variables in the numerator<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\sqrt{12}\\left(a\\right)+9[\/latex]<\/td>\r\n<td>\u00a0[latex]\\sqrt{a}+7[\/latex]<\/td>\r\n<td>\u00a0Variables under a root are not allowed in polynomials<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe basic building block of a polynomial is a <b>monomial<\/b>. When it is of the form [latex]a{x}^{m}[\/latex], where [latex]a[\/latex] is a constant and [latex]m[\/latex] is a whole number, it is called a monomial. A monomial is one term and can be a number, a variable, or the product of a number and variables with an exponent. The number part of the term is called the <b>coefficient<\/b>.\r\n\r\nExamples of monomials:\r\n<ul>\r\n \t<li>number, or coefficient: [latex]{6}[\/latex]<\/li>\r\n \t<li>variable: [latex]{x}[\/latex]<\/li>\r\n \t<li>product of coefficient and variable: [latex]{6x}[\/latex]<\/li>\r\n \t<li>product of coefficient and variable with an exponent: [latex]{6x}^{3}[\/latex]<\/li>\r\n<\/ul>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064120\/image003.jpg\" alt=\"The expression 6x to the power of 3. 6 is the coefficient, x is the variable, and the power of 3 is the exponent.\" width=\"183\" height=\"82\" \/>\r\n\r\nThe coefficient can be any real number, including [latex]0[\/latex]. The exponent of the variable must be a whole number [latex]\u20140, 1, 2, 3,[\/latex] and so on.\u00a0\u00a0The value of the exponent is the <b>degree<\/b> of the monomial. Remember that a variable that appears to have no exponent really has an exponent of [latex]1[\/latex]. And a monomial with no variable has a degree of \u00a0[latex]0[\/latex]. (Since\u00a0[latex]x^{0}[\/latex]\u00a0has the value of 1 if [latex]x\\neq0[\/latex],\u00a0a number such as [latex]3[\/latex] could also be written [latex]3x^{0}[\/latex], if [latex]x\\neq0[\/latex]\u00a0as [latex]3x^{0}=3\\cdot1=3[\/latex].)\u00a0 A monomial cannot have a variable in the denominator or a negative exponent.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nIdentify the coefficient, variable, and degree\u00a0of the variable for the following monomial terms:\r\n1) [latex]9[\/latex]\r\n2) [latex]x[\/latex]\r\n3) [latex]\\displaystyle \\frac{3}{5}{{k}^{8}}[\/latex]\r\n\r\n[reveal-answer q=\"150661\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"150661\"]\r\n1) [latex]9[\/latex] is a constant so there is no coefficient or variable. Since there is no variable, we consider the degree to be [latex]0[\/latex].\r\n\r\n2) The variable is [latex]x[\/latex].\r\n\r\nThe exponent of [latex]x[\/latex] is [latex]1[\/latex]. [latex]x=x^{1}[\/latex], so the degree is [latex]1[\/latex].\r\n\r\nThe coefficient of [latex]x[\/latex] is [latex]1[\/latex]. [latex]x=1x^{1}[\/latex].\r\n\r\n3) The variable is [latex]k[\/latex].\r\n\r\nThe exponent of [latex]k[\/latex] is [latex]8[\/latex], so the degree is [latex]8[\/latex].\r\n\r\nThe coefficient of [latex]k^{8}[\/latex]\u00a0is [latex] \\displaystyle \\frac{3}{5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 id=\"title2\">Evaluate a polynomial for given values of the variable<\/h2>\r\nPreviously we evaluated expressions by \"plugging in\" numbers for variables. Since polynomials are expressions, we'll follow the same procedures to evaluate polynomials\u2014substitute the given value for the variable into the polynomial, and then simplify.\u00a0 To evaluate an expression for a value of the variable, you substitute the value for the variable <i>every time<\/i> it appears. Then use the order of operations to find the resulting value for the expression.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nEvaluate [latex]3{x}^{2}-9x+7[\/latex] when\r\n\r\n1. [latex]x=3[\/latex]\r\n2. [latex]x=-1[\/latex]\r\n\r\nSolution\r\n<table id=\"eip-id1168468653511\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1. [latex]x=3[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3{x}^{2}-9x+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]3[\/latex] for [latex]x[\/latex]<\/td>\r\n<td>[latex]3{\\left(3\\right)}^{2}-9\\left(3\\right)+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify the expression with the exponent.<\/td>\r\n<td>[latex]3\\cdot 9 - 9\\left(3\\right)+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]27 - 27+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469859387\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2. [latex]x=-1[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3{x}^{2}-9x+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\u22121[\/latex] for [latex]x[\/latex]<\/td>\r\n<td>[latex]3{\\left(-1\\right)}^{2}-9\\left(-1\\right)+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify the expression with the exponent.<\/td>\r\n<td>[latex]3\\cdot 1 - 9\\left(-1\\right)+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]3+9+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]19[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nEvaluate [latex]3x^{2}-2x+1[\/latex] for [latex]x=-1[\/latex].\r\n\r\n[reveal-answer q=\"280466\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"280466\"]Substitute [latex]-1[\/latex] for each <i>x<\/i> in the polynomial.\r\n<p style=\"text-align: center;\">[latex]3\\left(-1\\right)^{2}-2\\left(-1\\right)+1[\/latex]<\/p>\r\nFollowing the order of operations, evaluate exponents first.\r\n<p style=\"text-align: center;\">[latex]3\\left(1\\right)-2\\left(-1\\right)+1[\/latex]<\/p>\r\nMultiply [latex]3[\/latex] times [latex]1[\/latex], and then multiply [latex]-2[\/latex] times [latex]-1[\/latex].\r\n<p style=\"text-align: center;\">[latex]3+\\left(-2\\right)\\left(-1\\right)+1[\/latex]<\/p>\r\nChange the subtraction to addition of the opposite.\r\n<p style=\"text-align: center;\">[latex]3+2+1[\/latex]<\/p>\r\nFind the sum.\r\n<h4>Answer<\/h4>\r\n[latex]3x^{2}-2x+1=6[\/latex], for [latex]x=-1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nEvaluate [latex] \\displaystyle -\\frac{2}{3}p^{4}+2p^{3}-p[\/latex] for [latex]p = 3[\/latex].\r\n\r\n[reveal-answer q=\"745542\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"745542\"]Substitute 3 for each <i>p<\/i> in the polynomial.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle -\\frac{2}{3}\\left(3\\right)^{4}+2\\left(3\\right)^{3}-3[\/latex]<\/p>\r\nFollowing the order of operations, evaluate exponents first and then multiply.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle -\\frac{2}{3}\\left(81\\right)+2\\left(27\\right)-3[\/latex]<\/p>\r\nAdd and then subtract to get [latex]-3[\/latex].\r\n<p style=\"text-align: center;\">[latex]-54 + 54 \u2013 3[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle -\\frac{2}{3}p^{4}+2p^{3}-p=-3[\/latex], for [latex]p = 3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146086[\/ohm_question]\r\n\r\n<\/div>\r\nThe following video presents more examples of evaluating a polynomial for a given value.\r\n\r\nhttps:\/\/youtu.be\/2EeFrgQP1hM\r\n\r\nThe following video provides another example of how to evaluate a quadratic polynomial for a negative number.\r\n\r\nhttps:\/\/youtu.be\/c7XkBD0fszc\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nThe polynomial [latex]-16{t}^{2}+300[\/latex] gives the height of an object [latex]t[\/latex] seconds after it is dropped from a [latex]300[\/latex] foot tall bridge. Find the height after [latex]t=3[\/latex] seconds.\r\n[reveal-answer q=\"237305\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"237305\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468766754\" class=\"unnumbered unstyled\" summary=\"The top line says negative 16 t squared plus 300. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-16t^2+300[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]3[\/latex] for [latex]t[\/latex]<\/td>\r\n<td>[latex]-16(\\color{red}{3})^2+300[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify the expression with the exponent.<\/td>\r\n<td>[latex]-16\\cdot{9}+300[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]-144+300[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]156[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146088[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Define polynomials<\/li>\n<li>Evaluate a polynomial for a given value<\/li>\n<\/ul>\n<\/div>\n<h2 id=\"title1\">Identify the terms, the coefficients, and the exponents of a polynomial<\/h2>\n<p>Polynomials are algebraic expressions that are created by combining numbers and variables using arithmetic operations such as addition, subtraction, multiplication, division, and exponentiation. A polynomial expression consists of the sum or difference of terms in which each term is a\u00a0real\u00a0number, a variable, or the product of a\u00a0real\u00a0number and variable(s) with a non-negative integer exponents. Non negative integers are\u00a0[latex]0, 1, 2, 3, 4[\/latex], &#8230;<\/p>\n<p>Polynomials are very useful in applications from science and engineering to business. You may see a resemblance between expressions, which we have been studying in this course, and polynomials. \u00a0Polynomials are a special sub-group of mathematical expressions and equations.<\/p>\n<p>The following table is intended to help you tell the difference between what is a polynomial and what is not.<\/p>\n<table>\n<thead>\n<tr>\n<td>IS a Polynomial<\/td>\n<td>Is NOT a Polynomial<\/td>\n<td>Because<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]2x^2-\\frac{1}{2}x -9[\/latex]<\/td>\n<td>[latex]\\frac{2}{x^{2}}+x[\/latex]<\/td>\n<td>Polynomials only have variables in the numerator<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\frac{y}{4}-y^3[\/latex]<\/td>\n<td>[latex]\\frac{2}{y}+4[\/latex]<\/td>\n<td>Polynomials only have variables in the numerator<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\sqrt{12}\\left(a\\right)+9[\/latex]<\/td>\n<td>\u00a0[latex]\\sqrt{a}+7[\/latex]<\/td>\n<td>\u00a0Variables under a root are not allowed in polynomials<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The basic building block of a polynomial is a <b>monomial<\/b>. When it is of the form [latex]a{x}^{m}[\/latex], where [latex]a[\/latex] is a constant and [latex]m[\/latex] is a whole number, it is called a monomial. A monomial is one term and can be a number, a variable, or the product of a number and variables with an exponent. The number part of the term is called the <b>coefficient<\/b>.<\/p>\n<p>Examples of monomials:<\/p>\n<ul>\n<li>number, or coefficient: [latex]{6}[\/latex]<\/li>\n<li>variable: [latex]{x}[\/latex]<\/li>\n<li>product of coefficient and variable: [latex]{6x}[\/latex]<\/li>\n<li>product of coefficient and variable with an exponent: [latex]{6x}^{3}[\/latex]<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064120\/image003.jpg\" alt=\"The expression 6x to the power of 3. 6 is the coefficient, x is the variable, and the power of 3 is the exponent.\" width=\"183\" height=\"82\" \/><\/p>\n<p>The coefficient can be any real number, including [latex]0[\/latex]. The exponent of the variable must be a whole number [latex]\u20140, 1, 2, 3,[\/latex] and so on.\u00a0\u00a0The value of the exponent is the <b>degree<\/b> of the monomial. Remember that a variable that appears to have no exponent really has an exponent of [latex]1[\/latex]. And a monomial with no variable has a degree of \u00a0[latex]0[\/latex]. (Since\u00a0[latex]x^{0}[\/latex]\u00a0has the value of 1 if [latex]x\\neq0[\/latex],\u00a0a number such as [latex]3[\/latex] could also be written [latex]3x^{0}[\/latex], if [latex]x\\neq0[\/latex]\u00a0as [latex]3x^{0}=3\\cdot1=3[\/latex].)\u00a0 A monomial cannot have a variable in the denominator or a negative exponent.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Identify the coefficient, variable, and degree\u00a0of the variable for the following monomial terms:<br \/>\n1) [latex]9[\/latex]<br \/>\n2) [latex]x[\/latex]<br \/>\n3) [latex]\\displaystyle \\frac{3}{5}{{k}^{8}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q150661\">Show Solution<\/span><\/p>\n<div id=\"q150661\" class=\"hidden-answer\" style=\"display: none\">\n1) [latex]9[\/latex] is a constant so there is no coefficient or variable. Since there is no variable, we consider the degree to be [latex]0[\/latex].<\/p>\n<p>2) The variable is [latex]x[\/latex].<\/p>\n<p>The exponent of [latex]x[\/latex] is [latex]1[\/latex]. [latex]x=x^{1}[\/latex], so the degree is [latex]1[\/latex].<\/p>\n<p>The coefficient of [latex]x[\/latex] is [latex]1[\/latex]. [latex]x=1x^{1}[\/latex].<\/p>\n<p>3) The variable is [latex]k[\/latex].<\/p>\n<p>The exponent of [latex]k[\/latex] is [latex]8[\/latex], so the degree is [latex]8[\/latex].<\/p>\n<p>The coefficient of [latex]k^{8}[\/latex]\u00a0is [latex]\\displaystyle \\frac{3}{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2 id=\"title2\">Evaluate a polynomial for given values of the variable<\/h2>\n<p>Previously we evaluated expressions by &#8220;plugging in&#8221; numbers for variables. Since polynomials are expressions, we&#8217;ll follow the same procedures to evaluate polynomials\u2014substitute the given value for the variable into the polynomial, and then simplify.\u00a0 To evaluate an expression for a value of the variable, you substitute the value for the variable <i>every time<\/i> it appears. Then use the order of operations to find the resulting value for the expression.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Evaluate [latex]3{x}^{2}-9x+7[\/latex] when<\/p>\n<p>1. [latex]x=3[\/latex]<br \/>\n2. [latex]x=-1[\/latex]<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168468653511\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1. [latex]x=3[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]3{x}^{2}-9x+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]3[\/latex] for [latex]x[\/latex]<\/td>\n<td>[latex]3{\\left(3\\right)}^{2}-9\\left(3\\right)+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the expression with the exponent.<\/td>\n<td>[latex]3\\cdot 9 - 9\\left(3\\right)+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]27 - 27+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]7[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469859387\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2. [latex]x=-1[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]3{x}^{2}-9x+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\u22121[\/latex] for [latex]x[\/latex]<\/td>\n<td>[latex]3{\\left(-1\\right)}^{2}-9\\left(-1\\right)+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the expression with the exponent.<\/td>\n<td>[latex]3\\cdot 1 - 9\\left(-1\\right)+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]3+9+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]19[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]3x^{2}-2x+1[\/latex] for [latex]x=-1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q280466\">Show Solution<\/span><\/p>\n<div id=\"q280466\" class=\"hidden-answer\" style=\"display: none\">Substitute [latex]-1[\/latex] for each <i>x<\/i> in the polynomial.<\/p>\n<p style=\"text-align: center;\">[latex]3\\left(-1\\right)^{2}-2\\left(-1\\right)+1[\/latex]<\/p>\n<p>Following the order of operations, evaluate exponents first.<\/p>\n<p style=\"text-align: center;\">[latex]3\\left(1\\right)-2\\left(-1\\right)+1[\/latex]<\/p>\n<p>Multiply [latex]3[\/latex] times [latex]1[\/latex], and then multiply [latex]-2[\/latex] times [latex]-1[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]3+\\left(-2\\right)\\left(-1\\right)+1[\/latex]<\/p>\n<p>Change the subtraction to addition of the opposite.<\/p>\n<p style=\"text-align: center;\">[latex]3+2+1[\/latex]<\/p>\n<p>Find the sum.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]3x^{2}-2x+1=6[\/latex], for [latex]x=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]\\displaystyle -\\frac{2}{3}p^{4}+2p^{3}-p[\/latex] for [latex]p = 3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q745542\">Show Solution<\/span><\/p>\n<div id=\"q745542\" class=\"hidden-answer\" style=\"display: none\">Substitute 3 for each <i>p<\/i> in the polynomial.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle -\\frac{2}{3}\\left(3\\right)^{4}+2\\left(3\\right)^{3}-3[\/latex]<\/p>\n<p>Following the order of operations, evaluate exponents first and then multiply.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle -\\frac{2}{3}\\left(81\\right)+2\\left(27\\right)-3[\/latex]<\/p>\n<p>Add and then subtract to get [latex]-3[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]-54 + 54 \u2013 3[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle -\\frac{2}{3}p^{4}+2p^{3}-p=-3[\/latex], for [latex]p = 3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146086\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146086&theme=oea&iframe_resize_id=ohm146086&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The following video presents more examples of evaluating a polynomial for a given value.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Evaluate a Polynomial in One Variable\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/2EeFrgQP1hM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The following video provides another example of how to evaluate a quadratic polynomial for a negative number.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Evaluate a Quadratic Expression With a Negative Value\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/c7XkBD0fszc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>The polynomial [latex]-16{t}^{2}+300[\/latex] gives the height of an object [latex]t[\/latex] seconds after it is dropped from a [latex]300[\/latex] foot tall bridge. Find the height after [latex]t=3[\/latex] seconds.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q237305\">Show Solution<\/span><\/p>\n<div id=\"q237305\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468766754\" class=\"unnumbered unstyled\" summary=\"The top line says negative 16 t squared plus 300. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]-16t^2+300[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]3[\/latex] for [latex]t[\/latex]<\/td>\n<td>[latex]-16(\\color{red}{3})^2+300[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the expression with the exponent.<\/td>\n<td>[latex]-16\\cdot{9}+300[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]-144+300[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]156[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146088\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146088&theme=oea&iframe_resize_id=ohm146088&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-16253\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Determine if an Expression is a Polynomial. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/nPAqfuoSbPI\">https:\/\/youtu.be\/nPAqfuoSbPI<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Evaluate a Polynomial in One Variable. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/2EeFrgQP1hM\">https:\/\/youtu.be\/2EeFrgQP1hM<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169554,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Determine if an Expression is a Polynomial\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/nPAqfuoSbPI\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Evaluate a Polynomial in One Variable\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/2EeFrgQP1hM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\" 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