{"id":4138,"date":"2022-08-01T23:51:54","date_gmt":"2022-08-01T23:51:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=4138"},"modified":"2026-01-20T20:08:36","modified_gmt":"2026-01-20T20:08:36","slug":"3-2-1-algebraic-analysis-of-polynomial-functions-3","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/3-2-1-algebraic-analysis-of-polynomial-functions-3\/","title":{"raw":"3.2.1: Algebraic Analysis of Polynomial Functions","rendered":"3.2.1: Algebraic Analysis of Polynomial Functions"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Evaluate function values<\/li>\r\n \t<li>Describe the meaning of assigning a function a value<\/li>\r\n \t<li>Perform addition and subtraction of polynomials<\/li>\r\n \t<li>Perform multiplication of polynomials<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Function Values<\/h2>\r\nAs we have seen in previous chapters, to evaluate a function, we substitute a value for the independent variable in the function equation and simplify the arithmetic expression. This value is the function value corresponding to the value of the independent variable. For example, to evaluate the value of the function [latex]f(x)=x^3-2x^2+5x-14[\/latex] when [latex]x = 1[\/latex], we substitute [latex]x[\/latex] with the value 1 and simplify:\r\n<p style=\"text-align: center;\">\u00a0[latex] \\begin{aligned} f(x)&amp;=x^3-2x^2+5x-14\\\\f(1) &amp;= (1)^3 - 2(1)^2 + 5(1) - 14 \\\\f(1)&amp;=1-2+5-14 \\\\f(1)&amp;= -10\\end{aligned}[\/latex]<\/p>\r\nTherefore, the function value when [latex]x = 1[\/latex] is \u201310, or, [latex]f(1) = -10[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nFor [latex]g(x)=3x^3-x^2+2x-5[\/latex], determine the function value,\r\n\r\n1. when [latex]x=2[\/latex]\r\n\r\n2. when [latex]x=-1[\/latex]\r\n\r\n3. [latex]g(0)[\/latex]\r\n\r\n4. [latex]g\\left (-\\frac{1}{2}\\right )[\/latex]\r\n<h4>Solution<\/h4>\r\nSubstitute the [latex]x[\/latex]-value into the function and simplify.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n<p style=\"text-align: initial;\">1.<\/p>\r\n<p style=\"text-align: initial;\">[latex]\\begin{aligned}g(x)&amp;=3x^3-x^2+2x-5\\\\g(2)&amp;=3(2)^3-(2)^2+2(2)-5\\\\&amp;=24-4+4-5\\\\&amp;=19\\end{aligned}[\/latex]<\/p>\r\n<\/td>\r\n<td style=\"width: 50%;\">\r\n<p style=\"text-align: initial;\">2.<\/p>\r\n<p style=\"text-align: initial;\">[latex]\\begin{aligned}g(x)&amp;=3x^3-x^2+2x-5\\\\g(-1)&amp;=3(-1)^3-(-1)^2+2(-1)-5\\\\&amp;=-3-1-2-5\\\\&amp;=-11\\end{aligned}[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n<p style=\"text-align: initial;\">3.<\/p>\r\n<p style=\"text-align: initial;\">[latex]\\begin{aligned}g(x)&amp;=3x^3-x^2+2x-5\\\\g(0)&amp;=3(0)^3-(0)^2+2(0)-5\\\\&amp;=0-0+0-5\\\\&amp;=-5\\end{aligned}[\/latex]<\/p>\r\n<p style=\"text-align: initial;\"><\/p>\r\n<\/td>\r\n<td style=\"width: 50%;\">\r\n<p style=\"text-align: initial;\">4.<\/p>\r\n<p style=\"text-align: initial;\">[latex]\\begin{aligned}g(x)&amp;=3x^3-x^2+2x-5\\\\g\\left (-\\frac{1}{2}\\right )&amp;=3\\left (-\\frac{1}{2}\\right )^3-\\left (-\\frac{1}{2}\\right )^2+2\\left (-\\frac{1}{2}\\right )-5\\\\&amp;=-\\dfrac{3}{8}-\\dfrac{1}{4}-1-5\\\\&amp;=\\dfrac{-3-2-8-40}{8}\\\\&amp;=-\\dfrac{53}{8}\\end{aligned}[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nFor [latex]f(x)=-x^3-x^2+3x+7[\/latex], determine the function value,\r\n\r\n1. when [latex]x=2[\/latex]\r\n\r\n2. when [latex]x=-1[\/latex]\r\n\r\n3. [latex]f(0)[\/latex]\r\n\r\n4. [latex]f(-2)[\/latex]\r\n\r\n[reveal-answer q=\"hjm914\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm914\"]\r\n<ol>\r\n \t<li>[latex]f(2)=1[\/latex]<\/li>\r\n \t<li>[latex]f(-1)=2[\/latex]<\/li>\r\n \t<li>[latex]f(0)=7[\/latex]<\/li>\r\n \t<li>[latex]f(-2)=5[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>The Meaning of Assigning a Function a Value<\/h2>\r\nAssigning a polynomial function a value (e.g., [latex]f(x) = 2 [\/latex]) has several meanings depending on perspective. First, it means finding the value(s) in the domain that is (are) mapped onto this function value in the range. For example, figure 1 shows the mapping of [latex]x[\/latex]-values to [latex]f(x)=x^2[\/latex]. If [latex]f(x)=a[\/latex], we need to find the value in the domain that results in a value of [latex]a[\/latex] for [latex]f(x)[\/latex].\r\n\r\n[caption id=\"attachment_1668\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1668 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/20210204\/mapping-x%5E2-300x221.png\" alt=\"mapping of x^2\" width=\"300\" height=\"221\" \/> Figure 1.The meaning of assigning a function a value: mapping[\/caption]\r\n\r\nSecond, since a function is composed of ordered pairs [latex](x, y)[\/latex] (or points [latex](x, y) [\/latex] on the coordinate plane), it may be understood as finding the [latex]x[\/latex]-coordinate given the value of the [latex]y[\/latex]-coordinate on the coordinate plane. For example, figure 2 shows the graph of a polynomial [latex]y=f(x)[\/latex]. If [latex]f(x)=1[\/latex] then [latex]y=1[\/latex] and the corresponding points on the graph when [latex]y=1[\/latex] are (\u20133, 1) and (\u20131, 1). Consequently, when\u00a0[latex]f(x)=1[\/latex], [latex]x=\u20131,\\;\u20133[\/latex].\r\n\r\n[caption id=\"attachment_1667\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1667 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/20203724\/desmos-graph-48-300x300.png\" alt=\"(x,y) points on graph\" width=\"300\" height=\"300\" \/> FIgure 2.\u00a0The meaning of assigning a function a value: coordinate points on a graph[\/caption]\r\n\r\nThird, assigning a function a value, [latex]f(x) = 2[\/latex] for example, means finding the intersection point between the graph of the function and the horizontal line [latex]y = 2[\/latex].\u00a0Figure 3 shows the function [latex]f(x)=x^2-4x+5[\/latex] and the horizontal line [latex]y=2[\/latex]. The parabola and the line intersect at the points (1, 2) and (3, 2). This shows that when [latex]f(x)=2[\/latex], the corresponding [latex]x[\/latex]-values are 1 and 3.\r\n\r\n[caption id=\"attachment_1662\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1662 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/20191228\/Meaning-of-fx2-300x291.png\" alt=\"Graph of parabola intersecting with a horizontal line\" width=\"300\" height=\"291\" \/> Figure 3. The meaning of assigning a function a value: intersection of graph with horizontal line[\/caption]\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nUse Desmos to determine the value(s) of [latex]x[\/latex] when [latex]f(x)=-1[\/latex] for [latex]f(x)=x^2+6x+7[\/latex].\r\n<h4>Solution<\/h4>\r\nTo determine when\u00a0[latex]x^2+6x+7=-1[\/latex] we look for the intersection point(s) between the graphs\u00a0[latex]y=x^2+6x+7[\/latex] and [latex]y=-1[\/latex].\r\n\r\nUsing Desmos:\r\n<p style=\"text-align: center;\"><img class=\"aligncenter size-medium wp-image-1663\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/20192437\/desmos-graph-44-300x300.png\" alt=\"graph of parabola and a line\" width=\"300\" height=\"300\" \/><\/p>\r\nFrom the graph, the intersection points of the parabola and the line are (\u20134, \u20131) and (\u20132, \u20131).\r\n\r\nSo,\u00a0[latex]f(x)=-1[\/latex] when [latex]x=-4,\\;-2[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nUse Desmos to determine the value(s) of [latex]x[\/latex] when [latex]f(x)=1[\/latex] for [latex]f(x)=-x^2-4x-2[\/latex].\r\n\r\n[reveal-answer q=\"hjm160\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm160\"]\r\n\r\n<img class=\"aligncenter size-medium wp-image-1664\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/20193043\/desmos-graph-47-300x300.png\" alt=\"parabola meets line\" width=\"300\" height=\"300\" \/>\r\n\r\n[latex]f(x)=-1[\/latex] when [latex]x=-3,\\;-1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Addition and Subtraction<\/h2>\r\nWe can add and subtract polynomial functions by combining like terms. For example, [latex]5{x}^{2}[\/latex] and [latex]-2{x}^{2}[\/latex] are like terms and can be added to get [latex]3{x}^{2}[\/latex], but [latex]3x[\/latex] and [latex]3{x}^{2}[\/latex] are not like terms, since they have different exponents, therefore, they cannot be added.\r\n<div class=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-134\" class=\"standard post-134 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nFind the sum of the polynomials, [latex]f(x)=12{x}^{2}+9x - 21[\/latex] and [latex]g(x)=4{x}^{3}+8{x}^{2}-5x+20[\/latex]\r\n<h4>Solution<\/h4>\r\n<p style=\"text-align: left;\">[latex]\\begin{aligned}f(x)+g(x)&amp;=\\left(12{x}^{2}+9x-21\\right)+\\left(4{x}^{3}+8{x}^{2}-5x+20\\right)\\\\&amp;=4{x}^{3}+\\left(12{x}^{2}+8{x}^{2}\\right)+\\left(9x - 5x\\right)+\\left(-21+20\\right) \\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ &amp;=4{x}^{3}+20{x}^{2}+4x - 1\\hfill &amp; \\text{Simplify}.\\hfill \\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nWhen we subtract polynomials, we need to pay attention to the sign of the terms we are combining. In the following example we will show how to distribute the negative sign to each term of a polynomial that is being subtracted from another.\r\n<div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nFind the difference: [latex]f(x)-g(x)[\/latex] when\u00a0[latex]7{x}^{4}-{x}^{2}+6x+1[\/latex] and\u00a0[latex]g(x)=5{x}^{3}-2{x}^{2}+3x+2[\/latex]\r\n<h4>Solution<\/h4>\r\n<p style=\"text-align: left;\">[latex]\\begin{aligned}f(x)-g(x)&amp;=\\left(7{x}^{4}-{x}^{2}+6x+1\\right)-\\left(5{x}^{3}-2{x}^{2}+3x+2\\right)\\\\&amp;=7{x}^4-{x}^2+6x+1-5{x}^3+2{x}^{2}-3x-2\\text{ }\\hfill &amp; \\text{Distribute \u20131 to all terms being subtracted}.\\hfill \\\\ &amp;=7{x}^{4}-5{x}^{3}+\\left(-{x}^{2}+2{x}^{2}\\right)+\\left(6x - 3x\\right)+\\left(1 - 2\\right)\\text{ }\\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ &amp;=7{x}^{4}-5{x}^{3}+{x}^{2}+3x - 1\\hfill &amp; \\text{Simplify}.\\hfill \\end{aligned}[\/latex]<\/p>\r\nNote that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\n<ol>\r\n \t<li>Find [latex]f(x)+g(x)[\/latex] when [latex]f(x)=3x^2-4x+2[\/latex] and [latex]g(x)=-3x^2-2x+1[\/latex]<\/li>\r\n \t<li>Find [latex]f(x)-g(x)[\/latex] when [latex]f(x)=5x^2-3x+2[\/latex] and [latex]g(x)=3x^2-3x-7[\/latex]<\/li>\r\n \t<li>Find [latex]f(x)+g(x)-h(x)[\/latex] when [latex]f(x)=3x^3-4x^2+4[\/latex], [latex]g(x)=2x^2-4x+7[\/latex] and [latex]h(x)=2x^3-8x^2-5x+3[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm002\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm002\"]\r\n<ol>\r\n \t<li>[latex]f(x)+g(x)=-6x+3[\/latex]<\/li>\r\n \t<li>[latex]f(x)-g(x)=2x^2+9[\/latex]<\/li>\r\n \t<li>[latex]f(x)+g(x)-h(x)=x^3+6x^2+x+8[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Multiplication<\/h2>\r\n<h3>Monomials<\/h3>\r\nThe most basic polynomial functions to multiply are <em><strong>monomials<\/strong><\/em>, i.e. polynomials with just one term. We use the properties of exponents to achieve this.\u00a0<span style=\"font-size: 1rem; text-align: initial;\">For example, [latex]x \\times x=x^2[\/latex]. This is because there are two\u00a0[latex]x[\/latex]s multiplied together, and we use the exponent 2 to specify the number of times the same base (e.g.,\u00a0[latex]x[\/latex]) is multiplied by itself. If we have\u00a0[latex]y \\times y \\times y \\times y[\/latex], we may write it as\u00a0[latex]y^4[\/latex]; [latex]y[\/latex] multiplied by itself 4 times. If we have\u00a0[latex] x \\times x^2 \\times x^5[\/latex]<\/span><span style=\"font-size: 1em;\">\u00a0we can expand each factor to get\u00a0<\/span><span style=\"font-size: 1em;\">[latex]x \\cdot (x \\cdot x) \\cdot (x \\cdot x \\cdot x \\cdot x \\cdot x)[\/latex].<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0<\/span><span style=\"font-size: 1em;\">Therefore, there are a total of eight [latex]x[\/latex]s that are multiplied together:\u00a0<span style=\"font-size: 1rem; text-align: initial;\">[latex] x \\times x^2 \\times x^5=x^8[\/latex]<\/span>.<\/span><span style=\"font-size: 1em;\">\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">Notice that we can add the exponents: <\/span><span style=\"font-size: 1rem; text-align: initial;\">[latex]1+2+5=8[\/latex]. This is an example of an exponential property called the multiplicative property of exponents.<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplicative property of exponents<\/h3>\r\nFor any [latex]x, m,[\/latex] and [latex]n\\in\\mathbb{R}[\/latex],\r\n<p style=\"text-align: center;\">[latex]x^m\\cdot x^n=x^{m+n}[\/latex]<\/p>\r\n\r\n<\/div>\r\nIn other words, to multiply exponential terms with the same base, we keep the common base and add the exponents.\r\n<div class=\"textbox examples\">\r\n<h3>Example 5<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>[latex]x^5\\cdot x^7[\/latex]<\/li>\r\n \t<li>[latex]y^4\\cdot y^3\\cdot y^0[\/latex]<\/li>\r\n \t<li>[latex]a^5\\cdot a^2\\cdot a^8\\cdot a^4[\/latex]<\/li>\r\n \t<li>[latex]x^4\\cdot y^2\\cdot x^5[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>[latex]x^5\\cdot x^7=x^{5+7}=x^{12}[\/latex]\u00a0 \u00a0Same base, so keep the base and add the exponents.<\/li>\r\n \t<li>[latex]y^4\\cdot y^3\\cdot y^0=y^{4+3+0}=y^7[\/latex]\u00a0 \u00a0Same base, so keep the base and add the exponents. Recall that [latex]y^0=1[\/latex] for all non-zero [latex]y[\/latex]-values. [latex]0^0[\/latex] is not defined.<\/li>\r\n \t<li>[latex]a^5\\cdot a^2\\cdot a^8\\cdot a^4=a^{5+2+8+4}=a^{19}[\/latex]\u00a0 \u00a0\u00a0Same base, so keep the base and add the exponents.<\/li>\r\n \t<li>[latex]x^4\\cdot y^2\\cdot x^5=x^{4+5}y^2=x^9y^2[\/latex]\u00a0 \u00a0 \u00a0Watch the bases. This cannot be simplified further since [latex]x[\/latex] and [latex]y[\/latex] are different bases.<\/li>\r\n<\/ol>\r\n<\/div>\r\nIf the exponential terms have a coefficient other than 1, we multiply the coefficients and multiply the variables.\r\n<div class=\"textbox examples\">\r\n<h3>Example 6<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>[latex]3x^3\\cdot 4x^4[\/latex]<\/li>\r\n \t<li>[latex]-5y^2\\left(3y^5\\right)\\left(2y^4\\right)[\/latex]<\/li>\r\n \t<li>[latex]7x^4\\cdot3y^7\\cdot7x^2\\cdot\\left(-3y^2\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>[latex]3x^3\\cdot 4x^4=(3\\cdot4)\\left(x^3\\cdot x^4\\right)=12x^7[\/latex]<\/li>\r\n \t<li>[latex]-5y^2\\left(3y^5\\right)\\left(2y^4\\right)=(-5\\cdot 3\\cdot 2)\\left(y^2 \\cdot y^5 \\cdot y^4 \\right) = -30y^{11} [\/latex]<\/li>\r\n \t<li>[latex]7x^4\\cdot 3y^7 \\cdot 7x^2 \\cdot \\left(-3y^2\\right)=\\left(7\\cdot 3\\cdot 7\\cdot(-3) \\right)\\left(x^{4+2}y^{7+2}\\right)=-441x^6y^9[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>[latex]x^5\\cdot x^9[\/latex]<\/li>\r\n \t<li>[latex]4y^6\\cdot5y^3[\/latex]<\/li>\r\n \t<li>[latex]-3x^3\\cdot5x^6\\cdot\\dfrac{4}{9}x[\/latex]<\/li>\r\n \t<li>[latex]-6y^5\\left(5x^2\\right)\\left(y^8\\right)(8x)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm659\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm659\"]\r\n<ol>\r\n \t<li>[latex]x^5\\cdot x^9=x^{14}[\/latex]<\/li>\r\n \t<li>[latex]4y^6\\cdot5y^3=20y^{9}[\/latex]<\/li>\r\n \t<li>[latex]-3x^3\\cdot5x^6\\cdot\\dfrac{4}{9}x=-\\dfrac{20}{3}x^{10}[\/latex]<\/li>\r\n \t<li>[latex]-6y^5\\left(5x^2\\right)\\left(y^8\\right)(8x)=-240x^3y^{13}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"wrap\">\r\n<h3>Monomial times a Polynomial<\/h3>\r\nTo multiply polynomial functions, we need a way to multiply over addition. The <em><strong>distributive property<\/strong><\/em>\u00a0allows us to multiply each term in the first polynomial by each term in the second polynomial. We then simplify by combining like terms.\r\n\r\nWe have used the distributive property in the past to multiply a number into an algebraic expression. For example, [latex]\\color{blue}{2}(x+7)=\\color{blue}{2}\\cdot x+\\color{blue}{2}\\cdot 7=2x+14[\/latex]. We say that we <em>distribute<\/em> the [latex]2[\/latex] to each term in the expression.\r\n\r\nWe can also apply the distributive property to terms. For example, [latex]\\color{blue}{3x}(2x-4)=\\color{blue}{3x}\\cdot 2x-\\color{blue}{3x}\\cdot 4=6x^2-12x[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>the distributive property of multiplication over addition<\/h3>\r\n<p style=\"text-align: center;\">[latex]a(b+c)=ab+ac[\/latex]<\/p>\r\n<p style=\"text-align: center;\">where [latex]a,\\;b,\\;[\/latex]and [latex]c[\/latex] are terms.<\/p>\r\n\r\n<\/div>\r\nThe following video shows examples of using the distributive property to find the product of\u00a0monomials (single terms) and polynomials.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/bwTmApTV_8o?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<div class=\"textbox examples\">\r\n<h3>Example 7<\/h3>\r\nFind [latex]f(x)\\cdot g(x)[\/latex] when [latex]f(x)=-3x^2[\/latex] and [latex]g(x)=4x^2-3x+2[\/latex].\r\n<h4>Solution<\/h4>\r\n[latex]\\begin{aligned}f(x)\\cdot g(x)&amp;=-3x^2 \\left (4x^2-3x+2 \\right )\\\\ &amp;=-3x^2(4x^2)-3x^2(-3x)-3x^2(2)\\\\&amp;=-12x^4+9x^3-6x^2\\end{aligned}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nFind [latex]f(x)\\cdot g(x)[\/latex] when [latex]f(x)=4x^3[\/latex] and [latex]g(x)=-3x^2+4x-5[\/latex].\r\n\r\n[reveal-answer q=\"hjm745\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm745\"]\r\n\r\n[latex]f(x)\\cdot g(x)=-12x^5+16x^4-20x^3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe now expand the distributive property to include polynomials. When multiplying polynomial functions, the distributive property allows us to multiply each term of the first polynomial function by each term of the second. We then add the products together and combine like terms to simplify.\r\n<div class=\"textbox shaded\">\r\n<h3>distributive property and polynomials<\/h3>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\left (a_nx^n+...+a_1x+a_0\\right )\\left (b_mx^m+...+b_1x+b_0\\right )\\\\=a_nx^n\\left (b_mx^m+...+b_1x+b_0\\right )+a_{n-1}x^{n-1}\\left (b_mx^m+...+b_1x+b_0\\right )+...+a_0\\left (b_mx^m+...+b_1x+b_0\\right )\\end{array}[\/latex]<\/p>\r\nEvery term in the first polynomial gets multiplied by every term in the second polynomial.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 8<\/h3>\r\nMultiply the polynomials:\u00a0 [latex]f(x)=3x+2[\/latex] and [latex]g(x)=3x^2+4x+2[\/latex]\r\n<h4>Solution<\/h4>\r\nEvery term in the binomial gets multiplied onto every term in the trinomial:\r\n\r\n[latex]\\begin{aligned}f(x)\\cdot g(x)&amp;=(\\color{blue}{3x}+\\color{red}{2})(3x^2+4x+2)\\\\&amp;=\\color{blue}{3x}(3x^2+4x+2)+\\color{red}{2}(3x^2+4x+2)\\\\&amp;=\\color{blue}{3x}(3x^2)+\\color{blue}{3x}\\color{black}{(4x)+}\\color{blue}{3x}(2)+\\color{red}{2}(3x^2)+\\color{red}{2}(4x)+\\color{red}{2}(2)\\\\&amp;=9x^3+12x^2+6x+6x^2+8x+4\\\\&amp;=9x^3+(12x^2+6x^2)+(6x+8x)+4\\\\&amp;=9x^3+18x^2+14x+4\\end{aligned}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n\r\n<span style=\"font-size: 1rem; orphans: 1; text-align: initial; widows: 2; display: inline !important;\">No matter how many terms are in the polynomials, we always multiply using the distributive property where every term in the first polynomial is multiplied onto every term in the second polynomial.<\/span>\r\n<div class=\"textbox examples\">\r\n<h3>Example 9<\/h3>\r\nMultiply the polynomial functions:\u00a0 [latex]f(x)=4x-3[\/latex] and [latex]g(x)=5x-6[\/latex]\r\n<h4>Solution<\/h4>\r\nEvery term in the first binomial gets multiplied onto every term in the second binomial:\r\n\r\n[latex]\\begin{aligned}f(x)\\cdot g(x)&amp;=(4x-3)(5x-6)\\\\&amp;=\\color{blue}{4x}(5x-6)\\color{red}{-3}(5x-6)\\\\&amp;=\\color{blue}{4x}(5x)+\\color{blue}{4x}(-6)\\color{red}{-3}(5x)\\color{red}{-3}(-6)\\\\&amp;=20x^2-24x-15x+18\\\\&amp;=20x^2-39x+18\\end{aligned}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 6<\/h3>\r\nMultiply the polynomial functions:\u00a0 [latex]f(x)=7x-4[\/latex] and [latex]g(x)=5x+3[\/latex]\r\n\r\n[reveal-answer q=\"hjm719\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm719\"]\r\n\r\n[latex]f(x)\\cdot g(x)=35x^2+x-12[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAs the number of terms in the polynomials increases, the more important it becomes to organize the terms after we distribute. One way to help organize terms, is to use vertical multiplication.\r\n<div class=\"textbox examples\">\r\n<h3>Example 10<\/h3>\r\nMultiply the functions: [latex]f(x)=3x+6[\/latex] and [latex]g(x)=5x^{2}+3x+10[\/latex]\r\n<h4>Solution<\/h4>\r\nSet up the problem in a vertical form, and begin by multiplying [latex]3x+6[\/latex] by [latex]+10[\/latex]. Place the products underneath, as shown.\u00a0Now multiply [latex]3x+6[\/latex] by 10, and place the two terms beneath the like terms.\r\n\r\n[latex]\\begin{array}{r}3x+\\,\\,\\,6\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}+\\,\\,3x+10}\\\\+30x+60\\,\\end{array}[\/latex]\r\n\r\nNow multiply [latex]3x+6[\/latex] by [latex]+3x[\/latex]. Notice that [latex]\\left(6\\right)\\left(3x\\right)=18x[\/latex]; since this term is like [latex]30x[\/latex], place it directly beneath it.\r\n\r\n[latex]\\begin{array}{r}3x\\,\\,\\,\\,\\,\\,+\\,\\,\\,6\\,\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}\\,\\,\\,\\,\\,\\,+3x\\,\\,\\,\\,\\,\\,+10}\\\\+30x\\,\\,\\,\\,\\,+60\\,\\,\\\\+9x^{2}\\,\\,\\,+18x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]\r\n\r\nFinally, multiply [latex]3x+6[\/latex] by [latex]5x^{2}[\/latex]. Notice that [latex]30x^{2}[\/latex]\u00a0is placed underneath [latex]9x^{2}[\/latex].\r\n\r\n[latex]\\begin{array}{r}3x\\,\\,\\,\\,\\,\\,+\\,\\,\\,6\\,\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}\\,\\,\\,\\,\\,\\,+3x\\,\\,\\,\\,\\,\\,+10}\\\\+30x\\,\\,\\,\\,\\,+60\\,\\,\\\\+9x^{2}\\,\\,\\,+18x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\underline{+15x^{3}+30x^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\end{array}[\/latex]\r\n\r\nNow add like terms.\r\n\r\n[latex]\\begin{array}{r}3x\\,\\,\\,\\,\\,\\,+\\,\\,\\,6\\,\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}\\,\\,\\,\\,\\,\\,+3x\\,\\,\\,\\,\\,\\,+10}\\\\+30x\\,\\,\\,\\,\\,+60\\,\\,\\\\+9x^{2}\\,\\,\\,+18x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\underline{+15x^{3}\\,\\,\\,\\,\\,\\,+30x^{2}\\;\\;\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\\\+15x^{3}\\,\\,\\,\\,\\,\\,+39x^{2}\\,\\,\\,\\,+48x\\,\\,\\,\\,\\,+60\\end{array}[\/latex]\r\n\r\nThe answer is [latex]f(x)\\cdot g(x)=15x^{3}+39x^{2}+48x+60[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 7<\/h3>\r\nFInd the product of [latex]f(x)=3x+2[\/latex] and [latex]g(x)=5x^2-3x+1[\/latex].\r\n\r\n[reveal-answer q=\"hjm427\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm427\"]\r\n\r\n[latex]f(x)\\cdot g(x)=15x^3+x^2-3x+2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAnother way to organize terms is do a hybrid version of the horizontal and vertical methods by writing like terms beneath one another as we distribute horizontally. This collects like terms in the same column so they are easily recognized and combined.\r\n<div class=\"textbox examples\">\r\n<h3>Example 11<\/h3>\r\nFind the product of\u00a0[latex]f(x)=2x+1[\/latex] and [latex]g(x)=3{x}^{2}-x+4[\/latex].\r\n<h4>Solution<\/h4>\r\n[latex]\\begin{array}{ll}f(x)\\cdot g(x)&amp;=\\left(2x+1\\right)\\left(3{x}^{2}-x+4\\right)&amp; \\\\ &amp;= \\color{blue}{2x}\\left(3{x}^{2}-x+4\\right)\\color{red}{+1}\\left(3{x}^{2}-x+4\\right)&amp;\\text{Distribute }\\color{blue}{2x}\\text{ and }\\color{red}{+1} \\\\&amp;= 6{x}^{3}-2{x}^{2}+8x\\\\&amp;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\,+3{x}^{2}-\\;\\,x+4&amp;\\text{Write like terms under one another}\\\\&amp;= 6{x}^{3}+{x}^{2}+7x+4&amp;\\text{Combine like terms}\\end{array}[\/latex]\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 8<\/h3>\r\nFind the product of\u00a0[latex]f(x)=4x-1[\/latex] and [latex]g(x)=2{x}^{2}-x+3[\/latex].\r\n\r\n[reveal-answer q=\"hjm657\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm657\"]\r\n\r\n[latex]f(x)\\cdot g(x)=8x^3-6x^2+13x-3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhichever method we choose to use, we will always end up with the same answer.\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">The following video shows more examples of multiplying polynomials.<\/span>\r\n\r\n<\/div>\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/bBKbldmlbqI?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\nWe can multiply polynomial functions using the distributive property no matter how many terms are in the polynomials.\r\n<div class=\"textbox examples\">\r\n<h3>Example 12<\/h3>\r\nFind the product of\u00a0[latex]f(x)=x^2+4x-1[\/latex] and [latex]g(x)=2{x}^{2}-3x+5[\/latex].\r\n<h4>Solution<\/h4>\r\n[latex]\\begin{array}{l}f(x)\\cdot g(x)&amp;=\\left (\\color{red}{x^2}\\color{blue}{+4x}\\color{green}{-1}\\right )\\left (2x^2-3x+5\\right )\\\\&amp;=\\color{red}{x^2}\\left (2x^2-3x+5\\right )\\color{blue}{+4x}\\left (2x^2-3x+5\\right )\\color{green}{-1}\\left (2x^2-3x+5\\right )\\\\&amp;=2x^4-3x^3+5x^2 \\\\&amp;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; +8x^3-12x^2+20x\\\\&amp;\\underline{\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;-2x^2\\,+\\;\\,3x-5}\\\\&amp;=2x^4+5x^3-9x^2+23x-5\\end{array}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 9<\/h3>\r\nFind the product of\u00a0[latex]f(x)=3x^2+x-2[\/latex] and [latex]g(x)=x^2-5x+4[\/latex].\r\n\r\n[reveal-answer q=\"hjm991\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm991\"]\r\n\r\n[latex]f(x)\\cdot g(x)=3x^4-14x^3+5x^2+14x-8[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Evaluate function values<\/li>\n<li>Describe the meaning of assigning a function a value<\/li>\n<li>Perform addition and subtraction of polynomials<\/li>\n<li>Perform multiplication of polynomials<\/li>\n<\/ul>\n<\/div>\n<h2>Function Values<\/h2>\n<p>As we have seen in previous chapters, to evaluate a function, we substitute a value for the independent variable in the function equation and simplify the arithmetic expression. This value is the function value corresponding to the value of the independent variable. For example, to evaluate the value of the function [latex]f(x)=x^3-2x^2+5x-14[\/latex] when [latex]x = 1[\/latex], we substitute [latex]x[\/latex] with the value 1 and simplify:<\/p>\n<p style=\"text-align: center;\">\u00a0[latex]\\begin{aligned} f(x)&=x^3-2x^2+5x-14\\\\f(1) &= (1)^3 - 2(1)^2 + 5(1) - 14 \\\\f(1)&=1-2+5-14 \\\\f(1)&= -10\\end{aligned}[\/latex]<\/p>\n<p>Therefore, the function value when [latex]x = 1[\/latex] is \u201310, or, [latex]f(1) = -10[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>For [latex]g(x)=3x^3-x^2+2x-5[\/latex], determine the function value,<\/p>\n<p>1. when [latex]x=2[\/latex]<\/p>\n<p>2. when [latex]x=-1[\/latex]<\/p>\n<p>3. [latex]g(0)[\/latex]<\/p>\n<p>4. [latex]g\\left (-\\frac{1}{2}\\right )[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>Substitute the [latex]x[\/latex]-value into the function and simplify.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">\n<p style=\"text-align: initial;\">1.<\/p>\n<p style=\"text-align: initial;\">[latex]\\begin{aligned}g(x)&=3x^3-x^2+2x-5\\\\g(2)&=3(2)^3-(2)^2+2(2)-5\\\\&=24-4+4-5\\\\&=19\\end{aligned}[\/latex]<\/p>\n<\/td>\n<td style=\"width: 50%;\">\n<p style=\"text-align: initial;\">2.<\/p>\n<p style=\"text-align: initial;\">[latex]\\begin{aligned}g(x)&=3x^3-x^2+2x-5\\\\g(-1)&=3(-1)^3-(-1)^2+2(-1)-5\\\\&=-3-1-2-5\\\\&=-11\\end{aligned}[\/latex]<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<p style=\"text-align: initial;\">3.<\/p>\n<p style=\"text-align: initial;\">[latex]\\begin{aligned}g(x)&=3x^3-x^2+2x-5\\\\g(0)&=3(0)^3-(0)^2+2(0)-5\\\\&=0-0+0-5\\\\&=-5\\end{aligned}[\/latex]<\/p>\n<p style=\"text-align: initial;\">\n<\/td>\n<td style=\"width: 50%;\">\n<p style=\"text-align: initial;\">4.<\/p>\n<p style=\"text-align: initial;\">[latex]\\begin{aligned}g(x)&=3x^3-x^2+2x-5\\\\g\\left (-\\frac{1}{2}\\right )&=3\\left (-\\frac{1}{2}\\right )^3-\\left (-\\frac{1}{2}\\right )^2+2\\left (-\\frac{1}{2}\\right )-5\\\\&=-\\dfrac{3}{8}-\\dfrac{1}{4}-1-5\\\\&=\\dfrac{-3-2-8-40}{8}\\\\&=-\\dfrac{53}{8}\\end{aligned}[\/latex]<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>For [latex]f(x)=-x^3-x^2+3x+7[\/latex], determine the function value,<\/p>\n<p>1. when [latex]x=2[\/latex]<\/p>\n<p>2. when [latex]x=-1[\/latex]<\/p>\n<p>3. [latex]f(0)[\/latex]<\/p>\n<p>4. [latex]f(-2)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm914\">Show Answer<\/span><\/p>\n<div id=\"qhjm914\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]f(2)=1[\/latex]<\/li>\n<li>[latex]f(-1)=2[\/latex]<\/li>\n<li>[latex]f(0)=7[\/latex]<\/li>\n<li>[latex]f(-2)=5[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>The Meaning of Assigning a Function a Value<\/h2>\n<p>Assigning a polynomial function a value (e.g., [latex]f(x) = 2[\/latex]) has several meanings depending on perspective. First, it means finding the value(s) in the domain that is (are) mapped onto this function value in the range. For example, figure 1 shows the mapping of [latex]x[\/latex]-values to [latex]f(x)=x^2[\/latex]. If [latex]f(x)=a[\/latex], we need to find the value in the domain that results in a value of [latex]a[\/latex] for [latex]f(x)[\/latex].<\/p>\n<div id=\"attachment_1668\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1668\" class=\"wp-image-1668 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/20210204\/mapping-x%5E2-300x221.png\" alt=\"mapping of x^2\" width=\"300\" height=\"221\" \/><\/p>\n<p id=\"caption-attachment-1668\" class=\"wp-caption-text\">Figure 1.The meaning of assigning a function a value: mapping<\/p>\n<\/div>\n<p>Second, since a function is composed of ordered pairs [latex](x, y)[\/latex] (or points [latex](x, y)[\/latex] on the coordinate plane), it may be understood as finding the [latex]x[\/latex]-coordinate given the value of the [latex]y[\/latex]-coordinate on the coordinate plane. For example, figure 2 shows the graph of a polynomial [latex]y=f(x)[\/latex]. If [latex]f(x)=1[\/latex] then [latex]y=1[\/latex] and the corresponding points on the graph when [latex]y=1[\/latex] are (\u20133, 1) and (\u20131, 1). Consequently, when\u00a0[latex]f(x)=1[\/latex], [latex]x=\u20131,\\;\u20133[\/latex].<\/p>\n<div id=\"attachment_1667\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1667\" class=\"wp-image-1667 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/20203724\/desmos-graph-48-300x300.png\" alt=\"(x,y) points on graph\" width=\"300\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-1667\" class=\"wp-caption-text\">FIgure 2.\u00a0The meaning of assigning a function a value: coordinate points on a graph<\/p>\n<\/div>\n<p>Third, assigning a function a value, [latex]f(x) = 2[\/latex] for example, means finding the intersection point between the graph of the function and the horizontal line [latex]y = 2[\/latex].\u00a0Figure 3 shows the function [latex]f(x)=x^2-4x+5[\/latex] and the horizontal line [latex]y=2[\/latex]. The parabola and the line intersect at the points (1, 2) and (3, 2). This shows that when [latex]f(x)=2[\/latex], the corresponding [latex]x[\/latex]-values are 1 and 3.<\/p>\n<div id=\"attachment_1662\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1662\" class=\"wp-image-1662 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/20191228\/Meaning-of-fx2-300x291.png\" alt=\"Graph of parabola intersecting with a horizontal line\" width=\"300\" height=\"291\" \/><\/p>\n<p id=\"caption-attachment-1662\" class=\"wp-caption-text\">Figure 3. The meaning of assigning a function a value: intersection of graph with horizontal line<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Use Desmos to determine the value(s) of [latex]x[\/latex] when [latex]f(x)=-1[\/latex] for [latex]f(x)=x^2+6x+7[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>To determine when\u00a0[latex]x^2+6x+7=-1[\/latex] we look for the intersection point(s) between the graphs\u00a0[latex]y=x^2+6x+7[\/latex] and [latex]y=-1[\/latex].<\/p>\n<p>Using Desmos:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-1663\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/20192437\/desmos-graph-44-300x300.png\" alt=\"graph of parabola and a line\" width=\"300\" height=\"300\" \/><\/p>\n<p>From the graph, the intersection points of the parabola and the line are (\u20134, \u20131) and (\u20132, \u20131).<\/p>\n<p>So,\u00a0[latex]f(x)=-1[\/latex] when [latex]x=-4,\\;-2[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Use Desmos to determine the value(s) of [latex]x[\/latex] when [latex]f(x)=1[\/latex] for [latex]f(x)=-x^2-4x-2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm160\">Show Answer<\/span><\/p>\n<div id=\"qhjm160\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-1664\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/20193043\/desmos-graph-47-300x300.png\" alt=\"parabola meets line\" width=\"300\" height=\"300\" \/><\/p>\n<p>[latex]f(x)=-1[\/latex] when [latex]x=-3,\\;-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Addition and Subtraction<\/h2>\n<p>We can add and subtract polynomial functions by combining like terms. For example, [latex]5{x}^{2}[\/latex] and [latex]-2{x}^{2}[\/latex] are like terms and can be added to get [latex]3{x}^{2}[\/latex], but [latex]3x[\/latex] and [latex]3{x}^{2}[\/latex] are not like terms, since they have different exponents, therefore, they cannot be added.<\/p>\n<div class=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-134\" class=\"standard post-134 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Find the sum of the polynomials, [latex]f(x)=12{x}^{2}+9x - 21[\/latex] and [latex]g(x)=4{x}^{3}+8{x}^{2}-5x+20[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p style=\"text-align: left;\">[latex]\\begin{aligned}f(x)+g(x)&=\\left(12{x}^{2}+9x-21\\right)+\\left(4{x}^{3}+8{x}^{2}-5x+20\\right)\\\\&=4{x}^{3}+\\left(12{x}^{2}+8{x}^{2}\\right)+\\left(9x - 5x\\right)+\\left(-21+20\\right) \\hfill & \\text{Combine like terms}.\\hfill \\\\ &=4{x}^{3}+20{x}^{2}+4x - 1\\hfill & \\text{Simplify}.\\hfill \\end{aligned}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>When we subtract polynomials, we need to pay attention to the sign of the terms we are combining. In the following example we will show how to distribute the negative sign to each term of a polynomial that is being subtracted from another.<\/p>\n<div>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>Find the difference: [latex]f(x)-g(x)[\/latex] when\u00a0[latex]7{x}^{4}-{x}^{2}+6x+1[\/latex] and\u00a0[latex]g(x)=5{x}^{3}-2{x}^{2}+3x+2[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p style=\"text-align: left;\">[latex]\\begin{aligned}f(x)-g(x)&=\\left(7{x}^{4}-{x}^{2}+6x+1\\right)-\\left(5{x}^{3}-2{x}^{2}+3x+2\\right)\\\\&=7{x}^4-{x}^2+6x+1-5{x}^3+2{x}^{2}-3x-2\\text{ }\\hfill & \\text{Distribute \u20131 to all terms being subtracted}.\\hfill \\\\ &=7{x}^{4}-5{x}^{3}+\\left(-{x}^{2}+2{x}^{2}\\right)+\\left(6x - 3x\\right)+\\left(1 - 2\\right)\\text{ }\\hfill & \\text{Combine like terms}.\\hfill \\\\ &=7{x}^{4}-5{x}^{3}+{x}^{2}+3x - 1\\hfill & \\text{Simplify}.\\hfill \\end{aligned}[\/latex]<\/p>\n<p>Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<ol>\n<li>Find [latex]f(x)+g(x)[\/latex] when [latex]f(x)=3x^2-4x+2[\/latex] and [latex]g(x)=-3x^2-2x+1[\/latex]<\/li>\n<li>Find [latex]f(x)-g(x)[\/latex] when [latex]f(x)=5x^2-3x+2[\/latex] and [latex]g(x)=3x^2-3x-7[\/latex]<\/li>\n<li>Find [latex]f(x)+g(x)-h(x)[\/latex] when [latex]f(x)=3x^3-4x^2+4[\/latex], [latex]g(x)=2x^2-4x+7[\/latex] and [latex]h(x)=2x^3-8x^2-5x+3[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm002\">Show Answer<\/span><\/p>\n<div id=\"qhjm002\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]f(x)+g(x)=-6x+3[\/latex]<\/li>\n<li>[latex]f(x)-g(x)=2x^2+9[\/latex]<\/li>\n<li>[latex]f(x)+g(x)-h(x)=x^3+6x^2+x+8[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Multiplication<\/h2>\n<h3>Monomials<\/h3>\n<p>The most basic polynomial functions to multiply are <em><strong>monomials<\/strong><\/em>, i.e. polynomials with just one term. We use the properties of exponents to achieve this.\u00a0<span style=\"font-size: 1rem; text-align: initial;\">For example, [latex]x \\times x=x^2[\/latex]. This is because there are two\u00a0[latex]x[\/latex]s multiplied together, and we use the exponent 2 to specify the number of times the same base (e.g.,\u00a0[latex]x[\/latex]) is multiplied by itself. If we have\u00a0[latex]y \\times y \\times y \\times y[\/latex], we may write it as\u00a0[latex]y^4[\/latex]; [latex]y[\/latex] multiplied by itself 4 times. If we have\u00a0[latex]x \\times x^2 \\times x^5[\/latex]<\/span><span style=\"font-size: 1em;\">\u00a0we can expand each factor to get\u00a0<\/span><span style=\"font-size: 1em;\">[latex]x \\cdot (x \\cdot x) \\cdot (x \\cdot x \\cdot x \\cdot x \\cdot x)[\/latex].<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0<\/span><span style=\"font-size: 1em;\">Therefore, there are a total of eight [latex]x[\/latex]s that are multiplied together:\u00a0<span style=\"font-size: 1rem; text-align: initial;\">[latex]x \\times x^2 \\times x^5=x^8[\/latex]<\/span>.<\/span><span style=\"font-size: 1em;\">\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">Notice that we can add the exponents: <\/span><span style=\"font-size: 1rem; text-align: initial;\">[latex]1+2+5=8[\/latex]. This is an example of an exponential property called the multiplicative property of exponents.<\/span><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Multiplicative property of exponents<\/h3>\n<p>For any [latex]x, m,[\/latex] and [latex]n\\in\\mathbb{R}[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]x^m\\cdot x^n=x^{m+n}[\/latex]<\/p>\n<\/div>\n<p>In other words, to multiply exponential terms with the same base, we keep the common base and add the exponents.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 5<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>[latex]x^5\\cdot x^7[\/latex]<\/li>\n<li>[latex]y^4\\cdot y^3\\cdot y^0[\/latex]<\/li>\n<li>[latex]a^5\\cdot a^2\\cdot a^8\\cdot a^4[\/latex]<\/li>\n<li>[latex]x^4\\cdot y^2\\cdot x^5[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>[latex]x^5\\cdot x^7=x^{5+7}=x^{12}[\/latex]\u00a0 \u00a0Same base, so keep the base and add the exponents.<\/li>\n<li>[latex]y^4\\cdot y^3\\cdot y^0=y^{4+3+0}=y^7[\/latex]\u00a0 \u00a0Same base, so keep the base and add the exponents. Recall that [latex]y^0=1[\/latex] for all non-zero [latex]y[\/latex]-values. [latex]0^0[\/latex] is not defined.<\/li>\n<li>[latex]a^5\\cdot a^2\\cdot a^8\\cdot a^4=a^{5+2+8+4}=a^{19}[\/latex]\u00a0 \u00a0\u00a0Same base, so keep the base and add the exponents.<\/li>\n<li>[latex]x^4\\cdot y^2\\cdot x^5=x^{4+5}y^2=x^9y^2[\/latex]\u00a0 \u00a0 \u00a0Watch the bases. This cannot be simplified further since [latex]x[\/latex] and [latex]y[\/latex] are different bases.<\/li>\n<\/ol>\n<\/div>\n<p>If the exponential terms have a coefficient other than 1, we multiply the coefficients and multiply the variables.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 6<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>[latex]3x^3\\cdot 4x^4[\/latex]<\/li>\n<li>[latex]-5y^2\\left(3y^5\\right)\\left(2y^4\\right)[\/latex]<\/li>\n<li>[latex]7x^4\\cdot3y^7\\cdot7x^2\\cdot\\left(-3y^2\\right)[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>[latex]3x^3\\cdot 4x^4=(3\\cdot4)\\left(x^3\\cdot x^4\\right)=12x^7[\/latex]<\/li>\n<li>[latex]-5y^2\\left(3y^5\\right)\\left(2y^4\\right)=(-5\\cdot 3\\cdot 2)\\left(y^2 \\cdot y^5 \\cdot y^4 \\right) = -30y^{11}[\/latex]<\/li>\n<li>[latex]7x^4\\cdot 3y^7 \\cdot 7x^2 \\cdot \\left(-3y^2\\right)=\\left(7\\cdot 3\\cdot 7\\cdot(-3) \\right)\\left(x^{4+2}y^{7+2}\\right)=-441x^6y^9[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>[latex]x^5\\cdot x^9[\/latex]<\/li>\n<li>[latex]4y^6\\cdot5y^3[\/latex]<\/li>\n<li>[latex]-3x^3\\cdot5x^6\\cdot\\dfrac{4}{9}x[\/latex]<\/li>\n<li>[latex]-6y^5\\left(5x^2\\right)\\left(y^8\\right)(8x)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm659\">Show Answer<\/span><\/p>\n<div id=\"qhjm659\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]x^5\\cdot x^9=x^{14}[\/latex]<\/li>\n<li>[latex]4y^6\\cdot5y^3=20y^{9}[\/latex]<\/li>\n<li>[latex]-3x^3\\cdot5x^6\\cdot\\dfrac{4}{9}x=-\\dfrac{20}{3}x^{10}[\/latex]<\/li>\n<li>[latex]-6y^5\\left(5x^2\\right)\\left(y^8\\right)(8x)=-240x^3y^{13}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"wrap\">\n<h3>Monomial times a Polynomial<\/h3>\n<p>To multiply polynomial functions, we need a way to multiply over addition. The <em><strong>distributive property<\/strong><\/em>\u00a0allows us to multiply each term in the first polynomial by each term in the second polynomial. We then simplify by combining like terms.<\/p>\n<p>We have used the distributive property in the past to multiply a number into an algebraic expression. For example, [latex]\\color{blue}{2}(x+7)=\\color{blue}{2}\\cdot x+\\color{blue}{2}\\cdot 7=2x+14[\/latex]. We say that we <em>distribute<\/em> the [latex]2[\/latex] to each term in the expression.<\/p>\n<p>We can also apply the distributive property to terms. For example, [latex]\\color{blue}{3x}(2x-4)=\\color{blue}{3x}\\cdot 2x-\\color{blue}{3x}\\cdot 4=6x^2-12x[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>the distributive property of multiplication over addition<\/h3>\n<p style=\"text-align: center;\">[latex]a(b+c)=ab+ac[\/latex]<\/p>\n<p style=\"text-align: center;\">where [latex]a,\\;b,\\;[\/latex]and [latex]c[\/latex] are terms.<\/p>\n<\/div>\n<p>The following video shows examples of using the distributive property to find the product of\u00a0monomials (single terms) and polynomials.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/bwTmApTV_8o?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox examples\">\n<h3>Example 7<\/h3>\n<p>Find [latex]f(x)\\cdot g(x)[\/latex] when [latex]f(x)=-3x^2[\/latex] and [latex]g(x)=4x^2-3x+2[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>[latex]\\begin{aligned}f(x)\\cdot g(x)&=-3x^2 \\left (4x^2-3x+2 \\right )\\\\ &=-3x^2(4x^2)-3x^2(-3x)-3x^2(2)\\\\&=-12x^4+9x^3-6x^2\\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>Find [latex]f(x)\\cdot g(x)[\/latex] when [latex]f(x)=4x^3[\/latex] and [latex]g(x)=-3x^2+4x-5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm745\">Show Answer<\/span><\/p>\n<div id=\"qhjm745\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f(x)\\cdot g(x)=-12x^5+16x^4-20x^3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We now expand the distributive property to include polynomials. When multiplying polynomial functions, the distributive property allows us to multiply each term of the first polynomial function by each term of the second. We then add the products together and combine like terms to simplify.<\/p>\n<div class=\"textbox shaded\">\n<h3>distributive property and polynomials<\/h3>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\left (a_nx^n+...+a_1x+a_0\\right )\\left (b_mx^m+...+b_1x+b_0\\right )\\\\=a_nx^n\\left (b_mx^m+...+b_1x+b_0\\right )+a_{n-1}x^{n-1}\\left (b_mx^m+...+b_1x+b_0\\right )+...+a_0\\left (b_mx^m+...+b_1x+b_0\\right )\\end{array}[\/latex]<\/p>\n<p>Every term in the first polynomial gets multiplied by every term in the second polynomial.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 8<\/h3>\n<p>Multiply the polynomials:\u00a0 [latex]f(x)=3x+2[\/latex] and [latex]g(x)=3x^2+4x+2[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>Every term in the binomial gets multiplied onto every term in the trinomial:<\/p>\n<p>[latex]\\begin{aligned}f(x)\\cdot g(x)&=(\\color{blue}{3x}+\\color{red}{2})(3x^2+4x+2)\\\\&=\\color{blue}{3x}(3x^2+4x+2)+\\color{red}{2}(3x^2+4x+2)\\\\&=\\color{blue}{3x}(3x^2)+\\color{blue}{3x}\\color{black}{(4x)+}\\color{blue}{3x}(2)+\\color{red}{2}(3x^2)+\\color{red}{2}(4x)+\\color{red}{2}(2)\\\\&=9x^3+12x^2+6x+6x^2+8x+4\\\\&=9x^3+(12x^2+6x^2)+(6x+8x)+4\\\\&=9x^3+18x^2+14x+4\\end{aligned}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<p><span style=\"font-size: 1rem; orphans: 1; text-align: initial; widows: 2; display: inline !important;\">No matter how many terms are in the polynomials, we always multiply using the distributive property where every term in the first polynomial is multiplied onto every term in the second polynomial.<\/span><\/p>\n<div class=\"textbox examples\">\n<h3>Example 9<\/h3>\n<p>Multiply the polynomial functions:\u00a0 [latex]f(x)=4x-3[\/latex] and [latex]g(x)=5x-6[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>Every term in the first binomial gets multiplied onto every term in the second binomial:<\/p>\n<p>[latex]\\begin{aligned}f(x)\\cdot g(x)&=(4x-3)(5x-6)\\\\&=\\color{blue}{4x}(5x-6)\\color{red}{-3}(5x-6)\\\\&=\\color{blue}{4x}(5x)+\\color{blue}{4x}(-6)\\color{red}{-3}(5x)\\color{red}{-3}(-6)\\\\&=20x^2-24x-15x+18\\\\&=20x^2-39x+18\\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 6<\/h3>\n<p>Multiply the polynomial functions:\u00a0 [latex]f(x)=7x-4[\/latex] and [latex]g(x)=5x+3[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm719\">Show Answer<\/span><\/p>\n<div id=\"qhjm719\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f(x)\\cdot g(x)=35x^2+x-12[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>As the number of terms in the polynomials increases, the more important it becomes to organize the terms after we distribute. One way to help organize terms, is to use vertical multiplication.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 10<\/h3>\n<p>Multiply the functions: [latex]f(x)=3x+6[\/latex] and [latex]g(x)=5x^{2}+3x+10[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>Set up the problem in a vertical form, and begin by multiplying [latex]3x+6[\/latex] by [latex]+10[\/latex]. Place the products underneath, as shown.\u00a0Now multiply [latex]3x+6[\/latex] by 10, and place the two terms beneath the like terms.<\/p>\n<p>[latex]\\begin{array}{r}3x+\\,\\,\\,6\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}+\\,\\,3x+10}\\\\+30x+60\\,\\end{array}[\/latex]<\/p>\n<p>Now multiply [latex]3x+6[\/latex] by [latex]+3x[\/latex]. Notice that [latex]\\left(6\\right)\\left(3x\\right)=18x[\/latex]; since this term is like [latex]30x[\/latex], place it directly beneath it.<\/p>\n<p>[latex]\\begin{array}{r}3x\\,\\,\\,\\,\\,\\,+\\,\\,\\,6\\,\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}\\,\\,\\,\\,\\,\\,+3x\\,\\,\\,\\,\\,\\,+10}\\\\+30x\\,\\,\\,\\,\\,+60\\,\\,\\\\+9x^{2}\\,\\,\\,+18x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Finally, multiply [latex]3x+6[\/latex] by [latex]5x^{2}[\/latex]. Notice that [latex]30x^{2}[\/latex]\u00a0is placed underneath [latex]9x^{2}[\/latex].<\/p>\n<p>[latex]\\begin{array}{r}3x\\,\\,\\,\\,\\,\\,+\\,\\,\\,6\\,\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}\\,\\,\\,\\,\\,\\,+3x\\,\\,\\,\\,\\,\\,+10}\\\\+30x\\,\\,\\,\\,\\,+60\\,\\,\\\\+9x^{2}\\,\\,\\,+18x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\underline{+15x^{3}+30x^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\end{array}[\/latex]<\/p>\n<p>Now add like terms.<\/p>\n<p>[latex]\\begin{array}{r}3x\\,\\,\\,\\,\\,\\,+\\,\\,\\,6\\,\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}\\,\\,\\,\\,\\,\\,+3x\\,\\,\\,\\,\\,\\,+10}\\\\+30x\\,\\,\\,\\,\\,+60\\,\\,\\\\+9x^{2}\\,\\,\\,+18x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\underline{+15x^{3}\\,\\,\\,\\,\\,\\,+30x^{2}\\;\\;\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\\\+15x^{3}\\,\\,\\,\\,\\,\\,+39x^{2}\\,\\,\\,\\,+48x\\,\\,\\,\\,\\,+60\\end{array}[\/latex]<\/p>\n<p>The answer is [latex]f(x)\\cdot g(x)=15x^{3}+39x^{2}+48x+60[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 7<\/h3>\n<p>FInd the product of [latex]f(x)=3x+2[\/latex] and [latex]g(x)=5x^2-3x+1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm427\">Show Answer<\/span><\/p>\n<div id=\"qhjm427\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f(x)\\cdot g(x)=15x^3+x^2-3x+2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Another way to organize terms is do a hybrid version of the horizontal and vertical methods by writing like terms beneath one another as we distribute horizontally. This collects like terms in the same column so they are easily recognized and combined.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 11<\/h3>\n<p>Find the product of\u00a0[latex]f(x)=2x+1[\/latex] and [latex]g(x)=3{x}^{2}-x+4[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>[latex]\\begin{array}{ll}f(x)\\cdot g(x)&=\\left(2x+1\\right)\\left(3{x}^{2}-x+4\\right)& \\\\ &= \\color{blue}{2x}\\left(3{x}^{2}-x+4\\right)\\color{red}{+1}\\left(3{x}^{2}-x+4\\right)&\\text{Distribute }\\color{blue}{2x}\\text{ and }\\color{red}{+1} \\\\&= 6{x}^{3}-2{x}^{2}+8x\\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\,+3{x}^{2}-\\;\\,x+4&\\text{Write like terms under one another}\\\\&= 6{x}^{3}+{x}^{2}+7x+4&\\text{Combine like terms}\\end{array}[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox tryit\">\n<h3>Try It 8<\/h3>\n<p>Find the product of\u00a0[latex]f(x)=4x-1[\/latex] and [latex]g(x)=2{x}^{2}-x+3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm657\">Show Answer<\/span><\/p>\n<div id=\"qhjm657\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f(x)\\cdot g(x)=8x^3-6x^2+13x-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Whichever method we choose to use, we will always end up with the same answer.<\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">The following video shows more examples of multiplying polynomials.<\/span><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/bBKbldmlbqI?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>We can multiply polynomial functions using the distributive property no matter how many terms are in the polynomials.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 12<\/h3>\n<p>Find the product of\u00a0[latex]f(x)=x^2+4x-1[\/latex] and [latex]g(x)=2{x}^{2}-3x+5[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>[latex]\\begin{array}{l}f(x)\\cdot g(x)&=\\left (\\color{red}{x^2}\\color{blue}{+4x}\\color{green}{-1}\\right )\\left (2x^2-3x+5\\right )\\\\&=\\color{red}{x^2}\\left (2x^2-3x+5\\right )\\color{blue}{+4x}\\left (2x^2-3x+5\\right )\\color{green}{-1}\\left (2x^2-3x+5\\right )\\\\&=2x^4-3x^3+5x^2 \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; +8x^3-12x^2+20x\\\\&\\underline{\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;-2x^2\\,+\\;\\,3x-5}\\\\&=2x^4+5x^3-9x^2+23x-5\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 9<\/h3>\n<p>Find the product of\u00a0[latex]f(x)=3x^2+x-2[\/latex] and [latex]g(x)=x^2-5x+4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm991\">Show Answer<\/span><\/p>\n<div id=\"qhjm991\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f(x)\\cdot g(x)=3x^4-14x^3+5x^2+14x-8[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\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-4138\">\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>Evaluate a Function, The Meaning of Assigning a Function a Value. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Figure 3. The Meaning of Assignment a Function a Value. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.desmos.com\/calculator\">http:\/\/www.desmos.com\/calculator<\/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>3.2.1 Algebraic Analysis of Polynomial Functions. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Multiplying Using the Distributive Property. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/bwTmApTV_8o\">https:\/\/youtu.be\/bwTmApTV_8o<\/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>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Figures 1 and 2; Examples and Try Its hjm914; hjm460; hjm002;. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <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. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/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>Ex: Polynomial Multiplication Involving Binomials and Trinomials. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/bBKbldmlbqI\">https:\/\/youtu.be\/bBKbldmlbqI<\/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":422608,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Evaluate a Function, The Meaning of Assigning a Function a Value\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Figure 3. The Meaning of Assignment a Function a Value\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"www.desmos.com\/calculator\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"3.2.1 Algebraic Analysis of Polynomial Functions\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Ex: Multiplying Using the Distributive Property\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/bwTmApTV_8o\",\"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\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Polynomial Multiplication Involving Binomials and Trinomials\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/bBKbldmlbqI\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Figures 1 and 2; Examples and Try Its hjm914; hjm460; hjm002;\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4138","chapter","type-chapter","status-publish","hentry"],"part":4124,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/4138","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/422608"}],"version-history":[{"count":10,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/4138\/revisions"}],"predecessor-version":[{"id":4835,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/4138\/revisions\/4835"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/4124"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/4138\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=4138"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=4138"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=4138"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=4138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}