{"id":2831,"date":"2022-06-20T17:55:51","date_gmt":"2022-06-20T17:55:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=2831"},"modified":"2026-01-18T03:20:48","modified_gmt":"2026-01-18T03:20:48","slug":"7-4-the-multiplication-and-division-of-rational-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/7-4-the-multiplication-and-division-of-rational-expressions\/","title":{"raw":"7.4: Multiplication and Division of Rational Functions","rendered":"7.4: Multiplication and Division of Rational 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=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-162\" class=\"standard post-162 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Simplify rational expressions<\/li>\r\n \t<li>Determine restrictions on a variable<\/li>\r\n \t<li>Perform multiplication and division of rational expressions<\/li>\r\n \t<li>Perform multiplication and division of rational functions<\/li>\r\n \t<li>Determine the domain of a product or quotient function<\/li>\r\n<\/ul>\r\n<\/div>\r\nEarlier we defined a <em><strong>rational expression<\/strong><\/em> as [latex]\\dfrac{P(x)}{Q(x)}[\/latex] where [latex]P(x)[\/latex] and [latex]Q(x)[\/latex] are polynomials. Just as we can multiply and divide fractions, we can multiply and divide\u00a0rational expressions. In fact, we use the same processes for multiplying and dividing rational expressions as we use for multiplying and dividing fractions. The process is the same even though the expressions look different.\r\n<h2>Simplifying Rational Expressions<\/h2>\r\nRational expressions that have common factors on the numerator and denominator can be simplified by cancelling out the common factors to 1. To determine if common factors exist, the numerator and denominators may have to be factored. However, we cannot divide by 0, so we must make sure to state any restrictions on the variable. These restrictions will appear when the denominator equals zero.\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nState the restrictions on the variable of the rational expression:\r\n\r\n1. [latex]\\dfrac{4x(2x-3)}{(x+4)(x-5)}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2. [latex]\\dfrac{7x+1}{5x(3x-2)}[\/latex]\r\n\r\n&nbsp;\r\n\r\n3. [latex]\\dfrac{4x^2-3x+5}{(x^2-9)(10x^2-7x-12)}[\/latex]\r\n<h4>Solution<\/h4>\r\nRestrictions occur in a rational expression when the denominator equals zero.\r\n\r\n1.\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}(x+4)(x-5)&amp;=0\\\\x+4=0\\text{ or }x-5&amp;=0\\\\x=-4,\\;x&amp;=5\\end{aligned}[\/latex]<\/p>\r\nConsequently, [latex]x\\neq -4,\\,5[\/latex]\r\n\r\n&nbsp;\r\n\r\n2.\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}5x(3x-2)&amp;=0\\\\5x=0\\text{ or }3x-2&amp;=0\\\\x=0,\\;x&amp;=\\frac{2}{3}\\end{aligned}[\/latex]<\/p>\r\nConsequently, [latex]x\\neq 0,\\,\\dfrac{2}{3}[\/latex]\r\n\r\n&nbsp;\r\n<p style=\"text-align: left;\">3. Often, we have to factor the denominator to determine the restrictions.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}(x^2-9)(10x^2-7x-12)&amp;=0\\\\(x-3)(x+3)(5x+4)(2x-3)&amp;=0\\\\x-3=0\\text{ or }x+3=0\\text{ or }5x+4=0\\text{ or }2x-3&amp;=0\\\\x=3,\\;x=-3,\\;x=-\\frac{4}{5},\\;x&amp;=\\frac{3}{2}\\end{aligned}[\/latex]<\/p>\r\nConsequently, [latex]x\\neq \\pm 3,\\,-\\dfrac{4}{5},\\,\\dfrac{3}{2}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nState the restrictions on the variable of the rational expression:\r\n\r\n1. [latex]\\dfrac{(2x-5)}{(x+2)(x-1)}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2. [latex]\\dfrac{3x+1}{5x^2(x-2)}[\/latex]\r\n\r\n&nbsp;\r\n\r\n3. [latex]\\dfrac{x^2}{(x^2-16)(x^2-7x-8)}[\/latex]\r\n\r\n[reveal-answer q=\"hjm775\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm775\"]\r\n<ol>\r\n \t<li>[latex]x\\neq -2,\\,1[\/latex]<\/li>\r\n \t<li>[latex]x\\neq 0,\\,2[\/latex]<\/li>\r\n \t<li>[latex]x\\neq \\pm 4,\\,-1,\\,8[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nState the restricted values and simplify.\r\n\r\n1. [latex]\\dfrac{4x^2}{8x^3+4x^2}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2. [latex]\\dfrac{x^2-9}{x^2-2x-15}[\/latex]\r\n\r\n&nbsp;\r\n\r\n3. [latex]\\dfrac{6x^2-7x-20}{2x^2+9x-35}[\/latex]\r\n<h4>Solution<\/h4>\r\nWe start by factoring the numerator and denominator to determine if there are any common factors. We make note of any restrictions on the variable, then cancel common factors.\r\n\r\n1.\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\dfrac{4x^2}{8x^3+4x^2}&amp;=\\dfrac{4x^2}{4x^2(2x+1)}&amp;&amp;\\text{Factor. Restrictions: }x\\neq 0,\\,-\\frac{1}{2}\\\\\\\\&amp;=\\dfrac{\\cancel{4x^2}}{\\cancel{4x^2}(2x+1)}&amp;&amp;\\text{Cancel: }\\dfrac{4x^2}{4x^2}=1\\\\\\\\&amp;=\\dfrac{1}{2x+1}\\end{aligned}[\/latex]<\/p>\r\n&nbsp;\r\n\r\n2.\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\dfrac{x^2-9}{x^2-2x-15}&amp;=\\dfrac{(x-3)(x+3)}{(x-5)(x+3)}&amp;&amp;\\text{Factor. Restrictions: }x\\neq 5,\\,-3\\\\\\\\&amp;=\\dfrac{(x-3)\\cancel{(x+3)}}{(x-5)\\cancel{(x+3)}}&amp;&amp;\\text{Cancel: }\\dfrac{x+3}{x+3}=1\\\\\\\\&amp;=\\dfrac{x-3}{x-5}\\end{aligned}[\/latex]<\/p>\r\n&nbsp;\r\n\r\n3.\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\dfrac{6x^2-7x-20}{2x^2+9x-35}&amp;=\\dfrac{(3x+4)(2x-5)}{(2x-5)(x+7)}&amp;&amp;\\text{Factor. Restrictions: }x\\neq \\frac{5}{2},\\,-7\\\\\\\\&amp;=\\dfrac{(3x+4)\\cancel{(2x-5)}}{\\cancel{(2x-5)}(x+7)}&amp;&amp;\\text{Cancel: }\\dfrac{2x-5}{2x-5}=1\\\\\\\\&amp;=\\dfrac{3x+4}{x+7}\\end{aligned}[\/latex]<\/p>\r\n<p style=\"text-align: center;\"><\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nState restricted values, then simplify.\r\n\r\n1. [latex]\\dfrac{9x^2}{18x^3+27x^2}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2. [latex]\\dfrac{x^2-16}{3x^2+11x-4}[\/latex]\r\n\r\n&nbsp;\r\n\r\n3. [latex]\\dfrac{6x^2+19x+10}{2x^2+13x+20}[\/latex]\r\n\r\n[reveal-answer q=\"hjm344\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm344\"]\r\n<ol>\r\n \t<li>[latex]\\dfrac{1}{2x+3}\\;\\;\\;\\;x\\neq 0,\\,-\\frac{3}{2}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{x-4}{3x-1}\\;\\;\\;\\;x\\neq -4,\\,\\frac{1}{3}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{3x+2}{x+4}\\;\\;\\;\\;x\\neq -4,\\,-\\frac{5}{2}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Multiplying Rational Expressions<\/h2>\r\nWe multiply rational expressions the same way we multiply fractions: [latex]\\dfrac{A}{B}\\cdot \\dfrac{C}{D}=\\dfrac{A\\cdot C}{B\\cdot D}[\/latex]. We can simplify by cancelling common factors, and we must state any restrictions on the variable.\r\n\r\n<\/div>\r\n<div class=\"bcc-box examples\">\r\n<h3>Example 3<\/h3>\r\nMultiply: [latex]\\displaystyle \\frac{5a^{2}}{14}\\cdot\\frac{7}{10a^{3}}[\/latex]\r\n<h4>Solution<\/h4>\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n<p style=\"text-align: center;\">[latex]\\begin {aligned}\\dfrac{5a^{2}}{14}\\cdot\\dfrac{7}{10a^{3}}&amp;=\\dfrac{5a^2\\cdot 7}{14\\cdot 10a^3}&amp;&amp;\\text{Multiply straight across}\\\\\\\\&amp;=\\dfrac{5\\cdot a^2\\cdot 7}{7\\cdot 2\\cdot 5\\cdot 2\\cdot a^2\\cdot a}&amp;&amp;\\text{Factor. Restrictions: }a\\neq 0\\\\\\\\&amp;=\\dfrac{\\cancel{5}\\cdot \\cancel{a^2}\\cdot \\cancel{7}}{\\cancel{7}\\cdot 2\\cdot\\cancel{5}\\cdot 2\\cdot\\cancel{a^2}\\cdot a}&amp;&amp;\\text{Cancel}\\\\\\\\&amp;=\\dfrac{1}{4a}\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box examples\">\r\n<h3>Example 4<\/h3>\r\nState the restrictions. Multiply:\u00a0 [latex]\\dfrac{{{a}^{2}}-a-2}{5a}\\cdot \\dfrac{10a}{a+1}[\/latex]. State the restrictions.\r\n<h4>Solution<\/h4>\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n<p style=\"text-align: left;\">Factor the numerators and denominators, then state the restrictions:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{{{a}^{2}}-a-2}{5a}\\cdot \\dfrac{10a}{a+1}=\\dfrac{\\left(a-2\\right)\\left(a+1\\right)}{5\\cdot{a}}\\cdot\\dfrac{5\\cdot2\\cdot{a}}{\\left(a+1\\right)}\\;\\;\\;\\;a\\neq 0,\\,-1[\/latex]<\/p>\r\nSimplify\u00a0common factors:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}&amp;\\dfrac{\\left(a-2\\right)\\cancel{\\left(a+1\\right)}}{\\cancel{5}\\cdot{\\cancel{a}}}\\cdot\\dfrac{\\cancel{5}\\cdot2\\cdot{\\cancel{a}}}{\\cancel{\\left(a+1\\right)}}\\\\\\\\&amp;=\\dfrac{a-2}{1}\\cdot\\dfrac{2}{1}\\end{aligned}[\/latex]<\/p>\r\nMultiply simplified rational expressions. This expression can be left with the numerator in factored form or multiplied out.\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}&amp;\\dfrac{\\left(a-2\\right)}{1}\\cdot\\dfrac{2}{1}\\\\\\\\&amp;=2\\left(a-2\\right)\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nState the restrictions on the variable, then multiply the rational expressions.\r\n\r\n1. [latex]\\dfrac{x+2}{x^2}\\cdot\\dfrac{5x}{x^2-4}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2. [latex]\\dfrac{4x-3}{x^2-2x+1}\\cdot\\dfrac{x-1}{16x^2-9}[\/latex]\r\n\r\n&nbsp;\r\n\r\n3. [latex]\\dfrac{x^2-3x-4}{4x}\\cdot \\dfrac{2x^2+2x}{x^2+7x+6}[\/latex]\r\n\r\n[reveal-answer q=\"hjm618\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm618\"]\r\n<ol>\r\n \t<li>[latex]\\dfrac{5}{x(x-2)}\\;\\;\\;\\;x\\neq \\pm2,\\,0[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{(x-1)(4x+3)}\\;\\;\\;\\;x\\neq 1,\\,\\pm\\frac{3}{4}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{(x-4)(x+1)}{2(x+6)}\\;\\;\\;\\;x\\neq 0,\\,-6,\\,-1[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>\u00a0Multiplying Rational Functions<\/h2>\r\nNow, we can apply the methods we learned multiplying rational expressions to multiply rational functions.\r\n<div class=\"textbox shaded\">\r\n<h3>The product function<\/h3>\r\n<p style=\"text-align: center;\">The product function [latex]\\left(f\\cdot g\\right)(x)=f(x)\\cdot g(x)[\/latex]<\/p>\r\n<p style=\"text-align: center;\">Its domain is all real numbers except the combined restrictions of [latex]f(x)[\/latex] and [latex]g(x)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"bcc-box examples\">\r\n<h3>Example 5<\/h3>\r\nState the domain of the product function, then multiply the two rational functions [latex]f(x)=\\dfrac{4x^5}{5x^2-125}[\/latex] and\u00a0[latex]g(x)=\\dfrac{x^2-3x-10}{20x^8}[\/latex].\r\n<h4>Solution<\/h4>\r\nBefore we perform the multiplication, we first need to discuss the domain of the product. The denominators of the two functions cannot be zero because division by zero is undefined. The domain of the product of two functions has restrictions equal to those of each function. In this case, [latex]5x^2-125=5(x^2-25)=5(x-5)(x+5)[\/latex], which when set equal to zero produces the restrictions [latex]x=\\pm 5[\/latex]. Also, [latex]20x^8=0[\/latex] when [latex]x=0[\/latex]. Consequently, the domain of [latex]\\left(f\\cdot g\\right)(x)=\\{x\\;|\\;x\\in \\mathbb{R},\\;x\\neq 0,\\,\\pm 5\\}[\/latex].\r\n\r\n&nbsp;\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}(f \\cdot g)(x)&amp;=f(x)\\cdot g(x)\\\\\\\\&amp;=\\dfrac{4x^5}{5x^2-125} \\times \\dfrac{x^2-3x-10}{20x^8}\\\\\\\\&amp;=\\dfrac{4x^5}{5(x^2-25)} \\times \\dfrac{(x-5)(x+2)}{20x^8}&amp;&amp;\\text{Factor}\\\\\\\\&amp;=\\dfrac{\\cancel{4}\\cancel{x^5}}{5(x+5)\\cancel{(x-5)}} \\times \\dfrac{\\cancel{(x-5)}(x+2)}{\\cancel{4} \\cdot 5 \\cdot \\cancel{x^5} \\cdot x^3}&amp;&amp;\\text{Cancel common factors to }1\\\\\\\\&amp;=\\dfrac{1}{5(x+5)} \\times \\dfrac{(x+2)}{5x^3}&amp;&amp;\\text{Multiply}\\\\\\\\&amp;= \\dfrac{(x+2)}{25x^3(x+5)}\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nState the domain of the product, then multiply the two rational functions [latex]f(x)=\\dfrac{4x^5}{5x^2-125}[\/latex] and\u00a0[latex]g(x)=\\dfrac{x^2-3x-10}{20x^8}[\/latex].\r\n\r\n[reveal-answer q=\"hjm203\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm203\"][latex]\\dfrac{x+2}{25x^3(x+5)}\\;\\;\\;x\\neq 0,\\,\\pm 5[\/latex]\r\n\r\nDomain of [latex]\\left(f\\cdot g\\right)(x)=\\{x\\;|\\;x\\in\\mathbb{R},\\;x\\neq 0,\\,\\pm 5[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Dividing Rational Expressions<\/h2>\r\nWe have seen that we multiply rational expressions as we multiply numerical fractions. It should come as no surprise that we also divide rational expressions the same way we divide numerical fractions. Specifically, to divide rational expressions, keep the first rational expression, change the division sign to multiplication, and then take the reciprocal of the second rational expression. [latex]\\dfrac{A}{B}\\div\\dfrac{C}{D}=\\dfrac{A}{B}\\cdot\\dfrac{D}{C}[\/latex].\r\n\r\nWe still need to think about restrictions on the variable, specifically, the variable values that would make either denominator equal zero. But there is a new consideration this time\u2014because we divide by multiplying by the reciprocal of the second rational expression, we also need to find the values that would make the\r\n<i>numerator <\/i>of that expression equal zero.\r\n<div class=\"bcc-box examples\">\r\n<h3>Example 6<\/h3>\r\nDivide:\u00a0 [latex]\\displaystyle\\frac{5x^{2}}{9}\\div\\frac{15x^{3}}{27}[\/latex]. State the restrictions on the variable.\r\n<h4>Solution<\/h4>\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\nTo determine the restricted values, first notice the two denominators, [latex]9[\/latex] and\u00a0[latex]27[\/latex], can never equal\u00a0[latex]0[\/latex]. This means there are no restrictions from those denominators.\r\n\r\nHowever, because [latex]15x^{3}[\/latex]\u00a0becomes the denominator in the reciprocal of [latex] \\displaystyle \\frac{15{{x}^{3}}}{27}[\/latex], we must find the values of [latex]x[\/latex] that would make [latex]15x^{3}=0[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}15x^{3}&amp;=0\\\\x&amp;=0\\end{aligned}[\/latex]<\/p>\r\nTherefore, [latex]x=0[\/latex] is a restricted value. i.e. [latex]x\\neq 0[\/latex].\r\n\r\n&nbsp;\r\n\r\nTo divide we rewrite division as multiplication by the reciprocal:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\dfrac{5x^{2}}{9}\\div\\dfrac{15x^{3}}{27}&amp;=\\dfrac{5x^{2}}{9}\\cdot\\frac{27}{15x^{3}}\\\\\\\\&amp;=\\dfrac{5\\cdot x^2\\cdot 9\\cdot 3}{9\\cdot 5 \\cdot 3\\cdot x^2\\cdot x}&amp;&amp;\\text{Factor}\\\\\\\\&amp;=\\dfrac{\\cancel{5}\\cdot\\cancel{x^2}\\cdot\\cancel{9}\\cdot\\cancel{3}}{\\cancel{9}\\cdot \\cancel{5} \\cdot \\cancel{3}\\cdot \\cancel{x^2}\\cdot x}&amp;&amp;\\text{Cancel}\\\\\\\\&amp;=\\dfrac{1}{x}&amp;&amp;\\text{Simplify}\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box examples\">\r\n<h3>Example 7<\/h3>\r\nDivide: [latex]\\displaystyle \\frac{3x^{2}}{x+2}\\div\\frac{6x^{4}}{\\left(x^{2}+5x+6\\right)}[\/latex]. State the restrictions.\r\n<h4>Solution<\/h4>\r\nWe can determine the restricted values as we work through the division.We will highlight the denominators where restrictions may lie in blue to make determining the restrictions on the domain more obvious.\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\dfrac{3x^2}{\\color{blue}{x+2}}\\div\\frac{6x^4}{\\left(\\color{blue}{x^2+5x+6}\\right)}&amp;=\\dfrac{3x^2}{x+2}\\cdot\\dfrac{\\left(\\color{blue}{x^2+5x+6}\\right)}{\\color{blue}{6x^4}}&amp;&amp;\\text{Factor}\\\\\\\\&amp;=\\dfrac{3x^2}{\\color{blue}{x+2}}\\cdot\\dfrac{\\color{blue}{(x+3)(x+2)}}{\\color{blue}{6x^4}}&amp;&amp;\\text{Determine restricted values: }x\\neq 0,\\,-2,\\,-3\\\\\\\\&amp;=\\dfrac{3\\cdot x^2}{x+2}\\cdot\\dfrac{(x+3)(x+2)}{2\\cdot 3\\cdot x^2\\cdot x^2}&amp;&amp;\\text{Factor}\\\\\\\\&amp;=\\dfrac{\\cancel{3}\\cdot\\cancel{x^2}}{\\cancel{(x+2)}}\\cdot\\dfrac{(x+3)\\cancel{(x+2)}}{2\\cdot\\cancel{3}\\cdot x^2\\cdot\\cancel{x^2}}&amp;&amp;\\text{Cancel}\\\\\\\\&amp;=\\dfrac{x+3}{2x^2}&amp;&amp;\\text{Simplify}\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\nNotice that once we rewrite the division as multiplication by a reciprocal, we follow the same process we use to multiply rational expressions.\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nDivide: [latex]\\dfrac{2x^3}{x^2-16}\\div\\dfrac{2x^3+4x^2}{x^2-2x-8}[\/latex]. State the restrictions on the variable.\r\n\r\n[reveal-answer q=\"hjm611\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm611\"]\r\n\r\n[latex]\\dfrac{x}{x+4}\\;\\;\\;\\;x\\neq \\pm 4,\\,0,\\,-2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Dividing Rational Functions<\/h2>\r\nNow, we can apply the methods we learned dividing rational expressions to divide rational functions.\r\n<div class=\"textbox shaded\">\r\n<h3>The quotient function<\/h3>\r\n<p style=\"text-align: center;\">The quotient function [latex]\\left(\\dfrac{f}{g}\\right)(x)=\\dfrac{f(x)}{g(x)}[\/latex].<\/p>\r\n<p style=\"text-align: center;\">Its domain is all real numbers except the combined restricted values of [latex]f(x)[\/latex] and\u00a0[latex]g(x)[\/latex], and the restricted values from the <em>numerator<\/em> of\u00a0[latex]g(x)[\/latex] since it becomes the denominator of the reciprocal.<\/p>\r\n\r\n<\/div>\r\n<div class=\"bcc-box examples\">\r\n<h3>Example 8<\/h3>\r\nDivide the rational functions [latex]f(x)=\\dfrac{x^2-6x+9}{4x^2-4x-24}[\/latex] and\u00a0[latex]g(x)=\\dfrac{5x^2-45}{20x^3}[\/latex]. State the domain of the quotient function.\r\n<h4>Solution<\/h4>\r\nBefore we perform the division, we first need to discuss the domain of the quotient. The denominators of the two functions cannot be zero because division by zero is undefined. The domain of the quotient of two functions has restrictions equal to those of each function, as well as any restrictions on the denominator of the reciprocal of the second function as we change division to multiplication of the reciprocal.\r\n\r\nWe will highlight the denominators where restrictions may lie in blue to make determining the restrictions on the domain more obvious.\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\left(\\dfrac{f}{g}\\right)(x)&amp;=f(x)\\div g(x)\\\\\\\\&amp;=\\dfrac{x^2-6x+9}{\\color{blue}{4x^2-4x-24}} \\div \\dfrac{5x^2-45}{\\color{blue}{20x^3}}\\\\\\\\&amp;=\\dfrac{x^2-6x+9}{\\color{blue}{4x^2-4x-24}} \\cdot \\dfrac{\\color{blue}{20x^3}}{\\color{blue}{5x^2-45}}&amp;&amp;\\text{Multiply by the reciprocal}\\\\\\\\&amp;=\\dfrac{x^2-6x+9}{\\color{blue}{4(x^2-x-6)}} \\cdot \\dfrac{\\color{blue}{20x^3}}{\\color{blue}{5(x^2-9)}}&amp;&amp;\\text{Factor}\\\\\\\\&amp;=\\dfrac{(x-3)(x-3)}{\\color{blue}{4(x-3)(x+2)}} \\cdot \\dfrac{\\color{blue}{20x^3}}{\\color{blue}{5(x-3)(x+3)}}&amp;&amp;\\text{Factor. Restrictions: }x\\neq \\pm 3,\\,-2,\\,0\\\\\\\\&amp;=\\dfrac{\\cancel{(x-3)}\\cancel{(x-3)}}{\\cancel{4}\\cancel{(x-3)}(x+2)} \\cdot \\dfrac{\\cancel{5} \\cdot \\cancel{4} \\cdot x^3}{\\cancel{5}(x+3)\\cancel{(x-3)}}&amp;&amp;\\text{Cancel}\\\\\\\\&amp;=\\dfrac{x^3}{(x+2)(x+3)}&amp;&amp;\\text{Simplify}\\end{aligned}[\/latex]<\/p>\r\n&nbsp;\r\n\r\nDomain of [latex]\\left(\\dfrac{f}{g}\\right)(x)=\\{x\\;|\\;x\\in\\mathbb{R},\\;x\\neq\\pm 3,\\,-2,\\,0\\}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 6<\/h3>\r\nDivide the rational functions [latex]f(x)=\\dfrac{2x^2+5x-12}{3x^2-2x}[\/latex] and\u00a0[latex]g(x)=\\dfrac{2x^2-x-3}{3x^2+x-2}[\/latex]. State the domain of the quotient function.\r\n\r\n[reveal-answer q=\"hjm104\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm104\"][latex]\\left(\\dfrac{f}{g}\\right)(x)=\\dfrac{x+4}{x}\\;\\;\\;x\\neq 0,\\,\\frac{2}{3},\\,\\frac{3}{2},\\,-1[\/latex]\r\n\r\nDomain of\u00a0[latex]\\left(\\dfrac{f}{g}\\right)(x)=\\{x\\;|\\;x\\in\\mathbb{R},\\;x\\neq 0,\\,\\frac{2}{3},\\,\\frac{3}{2},\\,-1\\}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Multiplying and Dividing Rational Functions<\/h2>\r\nThe techniques we have learned to multiply and divide rational functions can be used to extend the number of functions being multiplied or divided. Any time we need to divide, we multiply by the reciprocal.\r\n<div class=\"textbox examples\">\r\n<h3>Example 9<\/h3>\r\nSimplify: [latex]\\left(\\dfrac{f\\cdot g}{h}\\right)(x)[\/latex] when [latex]f(x)=\\dfrac{x^3+2x^2}{x+1}[\/latex],\u00a0[latex]g(x)=\\dfrac{x^2-2x-3}{x}[\/latex], and [latex]h(x)=\\dfrac{x^2-3x}{x^2+7x+10}[\/latex].\r\n\r\nState the domain of the product\/quotient function.\r\n<h4>Solution<\/h4>\r\nBlue highlights where restrictions on the domain are found.\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\left(\\dfrac{f\\cdot g}{h}\\right)(x)&amp;=\\dfrac{x^3+2x^2}{x+1}\\cdot\\dfrac{x^2-2x-3}{x}\\div\\dfrac{x^2-3x}{x^2+7x+10}\\\\\\\\&amp;=\\dfrac{x^3+2x^2}{\\color{blue}{x+1}}\\cdot\\dfrac{x^2-2x-3}{\\color{blue}{x}}\\cdot \\dfrac{\\color{blue}{x^2+7x+10}}{\\color{blue}{x^2-3x}}&amp;&amp;\\text{Write division as multiplication of the reciprocal}\\\\\\\\&amp;=\\dfrac{x^2(x+2)}{\\color{blue}{x+1}}\\cdot\\dfrac{(x-3)(x+1)}{\\color{blue}{x}}\\cdot \\dfrac{\\color{blue}{(x+5)(x+2)}}{\\color{blue}{x(x-3)}}&amp;&amp;\\text{Factor. Restrictions: }x\\neq -1,\\,0,\\,3,\\,-2,\\,-5\\\\\\\\&amp;=\\dfrac{\\cancel{x^2}(x+2)}{\\cancel{(x+1)}}\\cdot\\dfrac{\\cancel{(x-3)}\\cancel{(x+1)}}{\\cancel{x}}\\cdot \\dfrac{(x+5)(x+2)}{\\cancel{x}\\cancel{(x-3)}}&amp;&amp;\\text{Cancel}\\\\\\\\&amp;=(x+5)(x+2)^2\\end{aligned}[\/latex]<\/p>\r\n&nbsp;\r\n\r\nDomain of\u00a0[latex]\\left(\\dfrac{f\\cdot g}{h}\\right)(x)=\\{x\\;|\\;x\\in\\mathbb{R},\\;x\\neq -1,\\,0,\\,-2,\\,-5\\}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 7<\/h3>\r\nSimplify: [latex]\\left(\\dfrac{f\\cdot h}{g}\\right)(x)[\/latex] when [latex]f(x)=\\dfrac{x^3+2x^2}{x+1}[\/latex],\u00a0[latex]g(x)=\\dfrac{x^2-2x-3}{x}[\/latex], and [latex]h(x)=\\dfrac{x^2-3x}{x^2+7x+10}[\/latex].\r\n\r\nState the domain of the product\/quotient function.\r\n\r\n[reveal-answer q=\"hjm015\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm015\"][latex]\\left(\\dfrac{f\\cdot h}{g}\\right)(x)=\\dfrac{x^4}{(x+1)^2(x+5)}\\;\\;x\\neq -1,\\,0,\\,-2\\,-5[\/latex]\r\n\r\nDomain of\u00a0[latex]\\left(\\dfrac{f\\cdot h}{g}\\right)(x)={x\\;|\\;x\\in\\mathbb{R},\\;x\\neq -1,\\,0,\\,-2\\,-5}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\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=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-162\" class=\"standard post-162 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Simplify rational expressions<\/li>\n<li>Determine restrictions on a variable<\/li>\n<li>Perform multiplication and division of rational expressions<\/li>\n<li>Perform multiplication and division of rational functions<\/li>\n<li>Determine the domain of a product or quotient function<\/li>\n<\/ul>\n<\/div>\n<p>Earlier we defined a <em><strong>rational expression<\/strong><\/em> as [latex]\\dfrac{P(x)}{Q(x)}[\/latex] where [latex]P(x)[\/latex] and [latex]Q(x)[\/latex] are polynomials. Just as we can multiply and divide fractions, we can multiply and divide\u00a0rational expressions. In fact, we use the same processes for multiplying and dividing rational expressions as we use for multiplying and dividing fractions. The process is the same even though the expressions look different.<\/p>\n<h2>Simplifying Rational Expressions<\/h2>\n<p>Rational expressions that have common factors on the numerator and denominator can be simplified by cancelling out the common factors to 1. To determine if common factors exist, the numerator and denominators may have to be factored. However, we cannot divide by 0, so we must make sure to state any restrictions on the variable. These restrictions will appear when the denominator equals zero.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>State the restrictions on the variable of the rational expression:<\/p>\n<p>1. [latex]\\dfrac{4x(2x-3)}{(x+4)(x-5)}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2. [latex]\\dfrac{7x+1}{5x(3x-2)}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>3. [latex]\\dfrac{4x^2-3x+5}{(x^2-9)(10x^2-7x-12)}[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>Restrictions occur in a rational expression when the denominator equals zero.<\/p>\n<p>1.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}(x+4)(x-5)&=0\\\\x+4=0\\text{ or }x-5&=0\\\\x=-4,\\;x&=5\\end{aligned}[\/latex]<\/p>\n<p>Consequently, [latex]x\\neq -4,\\,5[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}5x(3x-2)&=0\\\\5x=0\\text{ or }3x-2&=0\\\\x=0,\\;x&=\\frac{2}{3}\\end{aligned}[\/latex]<\/p>\n<p>Consequently, [latex]x\\neq 0,\\,\\dfrac{2}{3}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left;\">3. Often, we have to factor the denominator to determine the restrictions.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}(x^2-9)(10x^2-7x-12)&=0\\\\(x-3)(x+3)(5x+4)(2x-3)&=0\\\\x-3=0\\text{ or }x+3=0\\text{ or }5x+4=0\\text{ or }2x-3&=0\\\\x=3,\\;x=-3,\\;x=-\\frac{4}{5},\\;x&=\\frac{3}{2}\\end{aligned}[\/latex]<\/p>\n<p>Consequently, [latex]x\\neq \\pm 3,\\,-\\dfrac{4}{5},\\,\\dfrac{3}{2}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>State the restrictions on the variable of the rational expression:<\/p>\n<p>1. [latex]\\dfrac{(2x-5)}{(x+2)(x-1)}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2. [latex]\\dfrac{3x+1}{5x^2(x-2)}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>3. [latex]\\dfrac{x^2}{(x^2-16)(x^2-7x-8)}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm775\">Show Answer<\/span><\/p>\n<div id=\"qhjm775\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]x\\neq -2,\\,1[\/latex]<\/li>\n<li>[latex]x\\neq 0,\\,2[\/latex]<\/li>\n<li>[latex]x\\neq \\pm 4,\\,-1,\\,8[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>State the restricted values and simplify.<\/p>\n<p>1. [latex]\\dfrac{4x^2}{8x^3+4x^2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2. [latex]\\dfrac{x^2-9}{x^2-2x-15}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>3. [latex]\\dfrac{6x^2-7x-20}{2x^2+9x-35}[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>We start by factoring the numerator and denominator to determine if there are any common factors. We make note of any restrictions on the variable, then cancel common factors.<\/p>\n<p>1.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\dfrac{4x^2}{8x^3+4x^2}&=\\dfrac{4x^2}{4x^2(2x+1)}&&\\text{Factor. Restrictions: }x\\neq 0,\\,-\\frac{1}{2}\\\\\\\\&=\\dfrac{\\cancel{4x^2}}{\\cancel{4x^2}(2x+1)}&&\\text{Cancel: }\\dfrac{4x^2}{4x^2}=1\\\\\\\\&=\\dfrac{1}{2x+1}\\end{aligned}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\dfrac{x^2-9}{x^2-2x-15}&=\\dfrac{(x-3)(x+3)}{(x-5)(x+3)}&&\\text{Factor. Restrictions: }x\\neq 5,\\,-3\\\\\\\\&=\\dfrac{(x-3)\\cancel{(x+3)}}{(x-5)\\cancel{(x+3)}}&&\\text{Cancel: }\\dfrac{x+3}{x+3}=1\\\\\\\\&=\\dfrac{x-3}{x-5}\\end{aligned}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>3.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\dfrac{6x^2-7x-20}{2x^2+9x-35}&=\\dfrac{(3x+4)(2x-5)}{(2x-5)(x+7)}&&\\text{Factor. Restrictions: }x\\neq \\frac{5}{2},\\,-7\\\\\\\\&=\\dfrac{(3x+4)\\cancel{(2x-5)}}{\\cancel{(2x-5)}(x+7)}&&\\text{Cancel: }\\dfrac{2x-5}{2x-5}=1\\\\\\\\&=\\dfrac{3x+4}{x+7}\\end{aligned}[\/latex]<\/p>\n<p style=\"text-align: center;\">\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>State restricted values, then simplify.<\/p>\n<p>1. [latex]\\dfrac{9x^2}{18x^3+27x^2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2. [latex]\\dfrac{x^2-16}{3x^2+11x-4}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>3. [latex]\\dfrac{6x^2+19x+10}{2x^2+13x+20}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm344\">Show Answer<\/span><\/p>\n<div id=\"qhjm344\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\dfrac{1}{2x+3}\\;\\;\\;\\;x\\neq 0,\\,-\\frac{3}{2}[\/latex]<\/li>\n<li>[latex]\\dfrac{x-4}{3x-1}\\;\\;\\;\\;x\\neq -4,\\,\\frac{1}{3}[\/latex]<\/li>\n<li>[latex]\\dfrac{3x+2}{x+4}\\;\\;\\;\\;x\\neq -4,\\,-\\frac{5}{2}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Multiplying Rational Expressions<\/h2>\n<p>We multiply rational expressions the same way we multiply fractions: [latex]\\dfrac{A}{B}\\cdot \\dfrac{C}{D}=\\dfrac{A\\cdot C}{B\\cdot D}[\/latex]. We can simplify by cancelling common factors, and we must state any restrictions on the variable.<\/p>\n<\/div>\n<div class=\"bcc-box examples\">\n<h3>Example 3<\/h3>\n<p>Multiply: [latex]\\displaystyle \\frac{5a^{2}}{14}\\cdot\\frac{7}{10a^{3}}[\/latex]<\/p>\n<h4>Solution<\/h4>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p style=\"text-align: center;\">[latex]\\begin {aligned}\\dfrac{5a^{2}}{14}\\cdot\\dfrac{7}{10a^{3}}&=\\dfrac{5a^2\\cdot 7}{14\\cdot 10a^3}&&\\text{Multiply straight across}\\\\\\\\&=\\dfrac{5\\cdot a^2\\cdot 7}{7\\cdot 2\\cdot 5\\cdot 2\\cdot a^2\\cdot a}&&\\text{Factor. Restrictions: }a\\neq 0\\\\\\\\&=\\dfrac{\\cancel{5}\\cdot \\cancel{a^2}\\cdot \\cancel{7}}{\\cancel{7}\\cdot 2\\cdot\\cancel{5}\\cdot 2\\cdot\\cancel{a^2}\\cdot a}&&\\text{Cancel}\\\\\\\\&=\\dfrac{1}{4a}\\end{aligned}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box examples\">\n<h3>Example 4<\/h3>\n<p>State the restrictions. Multiply:\u00a0 [latex]\\dfrac{{{a}^{2}}-a-2}{5a}\\cdot \\dfrac{10a}{a+1}[\/latex]. State the restrictions.<\/p>\n<h4>Solution<\/h4>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p style=\"text-align: left;\">Factor the numerators and denominators, then state the restrictions:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{{{a}^{2}}-a-2}{5a}\\cdot \\dfrac{10a}{a+1}=\\dfrac{\\left(a-2\\right)\\left(a+1\\right)}{5\\cdot{a}}\\cdot\\dfrac{5\\cdot2\\cdot{a}}{\\left(a+1\\right)}\\;\\;\\;\\;a\\neq 0,\\,-1[\/latex]<\/p>\n<p>Simplify\u00a0common factors:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}&\\dfrac{\\left(a-2\\right)\\cancel{\\left(a+1\\right)}}{\\cancel{5}\\cdot{\\cancel{a}}}\\cdot\\dfrac{\\cancel{5}\\cdot2\\cdot{\\cancel{a}}}{\\cancel{\\left(a+1\\right)}}\\\\\\\\&=\\dfrac{a-2}{1}\\cdot\\dfrac{2}{1}\\end{aligned}[\/latex]<\/p>\n<p>Multiply simplified rational expressions. This expression can be left with the numerator in factored form or multiplied out.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}&\\dfrac{\\left(a-2\\right)}{1}\\cdot\\dfrac{2}{1}\\\\\\\\&=2\\left(a-2\\right)\\end{aligned}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>State the restrictions on the variable, then multiply the rational expressions.<\/p>\n<p>1. [latex]\\dfrac{x+2}{x^2}\\cdot\\dfrac{5x}{x^2-4}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2. [latex]\\dfrac{4x-3}{x^2-2x+1}\\cdot\\dfrac{x-1}{16x^2-9}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>3. [latex]\\dfrac{x^2-3x-4}{4x}\\cdot \\dfrac{2x^2+2x}{x^2+7x+6}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm618\">Show Answer<\/span><\/p>\n<div id=\"qhjm618\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\dfrac{5}{x(x-2)}\\;\\;\\;\\;x\\neq \\pm2,\\,0[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{(x-1)(4x+3)}\\;\\;\\;\\;x\\neq 1,\\,\\pm\\frac{3}{4}[\/latex]<\/li>\n<li>[latex]\\dfrac{(x-4)(x+1)}{2(x+6)}\\;\\;\\;\\;x\\neq 0,\\,-6,\\,-1[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>\u00a0Multiplying Rational Functions<\/h2>\n<p>Now, we can apply the methods we learned multiplying rational expressions to multiply rational functions.<\/p>\n<div class=\"textbox shaded\">\n<h3>The product function<\/h3>\n<p style=\"text-align: center;\">The product function [latex]\\left(f\\cdot g\\right)(x)=f(x)\\cdot g(x)[\/latex]<\/p>\n<p style=\"text-align: center;\">Its domain is all real numbers except the combined restrictions of [latex]f(x)[\/latex] and [latex]g(x)[\/latex].<\/p>\n<\/div>\n<div class=\"bcc-box examples\">\n<h3>Example 5<\/h3>\n<p>State the domain of the product function, then multiply the two rational functions [latex]f(x)=\\dfrac{4x^5}{5x^2-125}[\/latex] and\u00a0[latex]g(x)=\\dfrac{x^2-3x-10}{20x^8}[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>Before we perform the multiplication, we first need to discuss the domain of the product. The denominators of the two functions cannot be zero because division by zero is undefined. The domain of the product of two functions has restrictions equal to those of each function. In this case, [latex]5x^2-125=5(x^2-25)=5(x-5)(x+5)[\/latex], which when set equal to zero produces the restrictions [latex]x=\\pm 5[\/latex]. Also, [latex]20x^8=0[\/latex] when [latex]x=0[\/latex]. Consequently, the domain of [latex]\\left(f\\cdot g\\right)(x)=\\{x\\;|\\;x\\in \\mathbb{R},\\;x\\neq 0,\\,\\pm 5\\}[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}(f \\cdot g)(x)&=f(x)\\cdot g(x)\\\\\\\\&=\\dfrac{4x^5}{5x^2-125} \\times \\dfrac{x^2-3x-10}{20x^8}\\\\\\\\&=\\dfrac{4x^5}{5(x^2-25)} \\times \\dfrac{(x-5)(x+2)}{20x^8}&&\\text{Factor}\\\\\\\\&=\\dfrac{\\cancel{4}\\cancel{x^5}}{5(x+5)\\cancel{(x-5)}} \\times \\dfrac{\\cancel{(x-5)}(x+2)}{\\cancel{4} \\cdot 5 \\cdot \\cancel{x^5} \\cdot x^3}&&\\text{Cancel common factors to }1\\\\\\\\&=\\dfrac{1}{5(x+5)} \\times \\dfrac{(x+2)}{5x^3}&&\\text{Multiply}\\\\\\\\&= \\dfrac{(x+2)}{25x^3(x+5)}\\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>State the domain of the product, then multiply the two rational functions [latex]f(x)=\\dfrac{4x^5}{5x^2-125}[\/latex] and\u00a0[latex]g(x)=\\dfrac{x^2-3x-10}{20x^8}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm203\">Show Answer<\/span><\/p>\n<div id=\"qhjm203\" class=\"hidden-answer\" style=\"display: none\">[latex]\\dfrac{x+2}{25x^3(x+5)}\\;\\;\\;x\\neq 0,\\,\\pm 5[\/latex]<\/p>\n<p>Domain of [latex][\/latex]\\left(f\\cdot g\\right)(x)=\\{x\\;|\\;x\\in\\mathbb{R},\\;x\\neq 0,\\,\\pm 5<\/p><\/div>\n<\/div>\n<\/div>\n<h2>Dividing Rational Expressions<\/h2>\n<p>We have seen that we multiply rational expressions as we multiply numerical fractions. It should come as no surprise that we also divide rational expressions the same way we divide numerical fractions. Specifically, to divide rational expressions, keep the first rational expression, change the division sign to multiplication, and then take the reciprocal of the second rational expression. [latex]\\dfrac{A}{B}\\div\\dfrac{C}{D}=\\dfrac{A}{B}\\cdot\\dfrac{D}{C}[\/latex].<\/p>\n<p>We still need to think about restrictions on the variable, specifically, the variable values that would make either denominator equal zero. But there is a new consideration this time\u2014because we divide by multiplying by the reciprocal of the second rational expression, we also need to find the values that would make the<br \/>\n<i>numerator <\/i>of that expression equal zero.<\/p>\n<div class=\"bcc-box examples\">\n<h3>Example 6<\/h3>\n<p>Divide:\u00a0 [latex]\\displaystyle\\frac{5x^{2}}{9}\\div\\frac{15x^{3}}{27}[\/latex]. State the restrictions on the variable.<\/p>\n<h4>Solution<\/h4>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p>To determine the restricted values, first notice the two denominators, [latex]9[\/latex] and\u00a0[latex]27[\/latex], can never equal\u00a0[latex]0[\/latex]. This means there are no restrictions from those denominators.<\/p>\n<p>However, because [latex]15x^{3}[\/latex]\u00a0becomes the denominator in the reciprocal of [latex]\\displaystyle \\frac{15{{x}^{3}}}{27}[\/latex], we must find the values of [latex]x[\/latex] that would make [latex]15x^{3}=0[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}15x^{3}&=0\\\\x&=0\\end{aligned}[\/latex]<\/p>\n<p>Therefore, [latex]x=0[\/latex] is a restricted value. i.e. [latex]x\\neq 0[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>To divide we rewrite division as multiplication by the reciprocal:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\dfrac{5x^{2}}{9}\\div\\dfrac{15x^{3}}{27}&=\\dfrac{5x^{2}}{9}\\cdot\\frac{27}{15x^{3}}\\\\\\\\&=\\dfrac{5\\cdot x^2\\cdot 9\\cdot 3}{9\\cdot 5 \\cdot 3\\cdot x^2\\cdot x}&&\\text{Factor}\\\\\\\\&=\\dfrac{\\cancel{5}\\cdot\\cancel{x^2}\\cdot\\cancel{9}\\cdot\\cancel{3}}{\\cancel{9}\\cdot \\cancel{5} \\cdot \\cancel{3}\\cdot \\cancel{x^2}\\cdot x}&&\\text{Cancel}\\\\\\\\&=\\dfrac{1}{x}&&\\text{Simplify}\\end{aligned}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"bcc-box examples\">\n<h3>Example 7<\/h3>\n<p>Divide: [latex]\\displaystyle \\frac{3x^{2}}{x+2}\\div\\frac{6x^{4}}{\\left(x^{2}+5x+6\\right)}[\/latex]. State the restrictions.<\/p>\n<h4>Solution<\/h4>\n<p>We can determine the restricted values as we work through the division.We will highlight the denominators where restrictions may lie in blue to make determining the restrictions on the domain more obvious.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\dfrac{3x^2}{\\color{blue}{x+2}}\\div\\frac{6x^4}{\\left(\\color{blue}{x^2+5x+6}\\right)}&=\\dfrac{3x^2}{x+2}\\cdot\\dfrac{\\left(\\color{blue}{x^2+5x+6}\\right)}{\\color{blue}{6x^4}}&&\\text{Factor}\\\\\\\\&=\\dfrac{3x^2}{\\color{blue}{x+2}}\\cdot\\dfrac{\\color{blue}{(x+3)(x+2)}}{\\color{blue}{6x^4}}&&\\text{Determine restricted values: }x\\neq 0,\\,-2,\\,-3\\\\\\\\&=\\dfrac{3\\cdot x^2}{x+2}\\cdot\\dfrac{(x+3)(x+2)}{2\\cdot 3\\cdot x^2\\cdot x^2}&&\\text{Factor}\\\\\\\\&=\\dfrac{\\cancel{3}\\cdot\\cancel{x^2}}{\\cancel{(x+2)}}\\cdot\\dfrac{(x+3)\\cancel{(x+2)}}{2\\cdot\\cancel{3}\\cdot x^2\\cdot\\cancel{x^2}}&&\\text{Cancel}\\\\\\\\&=\\dfrac{x+3}{2x^2}&&\\text{Simplify}\\end{aligned}[\/latex]<\/p>\n<\/div>\n<p>Notice that once we rewrite the division as multiplication by a reciprocal, we follow the same process we use to multiply rational expressions.<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>Divide: [latex]\\dfrac{2x^3}{x^2-16}\\div\\dfrac{2x^3+4x^2}{x^2-2x-8}[\/latex]. State the restrictions on the variable.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm611\">Show Answer<\/span><\/p>\n<div id=\"qhjm611\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{x}{x+4}\\;\\;\\;\\;x\\neq \\pm 4,\\,0,\\,-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Dividing Rational Functions<\/h2>\n<p>Now, we can apply the methods we learned dividing rational expressions to divide rational functions.<\/p>\n<div class=\"textbox shaded\">\n<h3>The quotient function<\/h3>\n<p style=\"text-align: center;\">The quotient function [latex]\\left(\\dfrac{f}{g}\\right)(x)=\\dfrac{f(x)}{g(x)}[\/latex].<\/p>\n<p style=\"text-align: center;\">Its domain is all real numbers except the combined restricted values of [latex]f(x)[\/latex] and\u00a0[latex]g(x)[\/latex], and the restricted values from the <em>numerator<\/em> of\u00a0[latex]g(x)[\/latex] since it becomes the denominator of the reciprocal.<\/p>\n<\/div>\n<div class=\"bcc-box examples\">\n<h3>Example 8<\/h3>\n<p>Divide the rational functions [latex]f(x)=\\dfrac{x^2-6x+9}{4x^2-4x-24}[\/latex] and\u00a0[latex]g(x)=\\dfrac{5x^2-45}{20x^3}[\/latex]. State the domain of the quotient function.<\/p>\n<h4>Solution<\/h4>\n<p>Before we perform the division, we first need to discuss the domain of the quotient. The denominators of the two functions cannot be zero because division by zero is undefined. The domain of the quotient of two functions has restrictions equal to those of each function, as well as any restrictions on the denominator of the reciprocal of the second function as we change division to multiplication of the reciprocal.<\/p>\n<p>We will highlight the denominators where restrictions may lie in blue to make determining the restrictions on the domain more obvious.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\left(\\dfrac{f}{g}\\right)(x)&=f(x)\\div g(x)\\\\\\\\&=\\dfrac{x^2-6x+9}{\\color{blue}{4x^2-4x-24}} \\div \\dfrac{5x^2-45}{\\color{blue}{20x^3}}\\\\\\\\&=\\dfrac{x^2-6x+9}{\\color{blue}{4x^2-4x-24}} \\cdot \\dfrac{\\color{blue}{20x^3}}{\\color{blue}{5x^2-45}}&&\\text{Multiply by the reciprocal}\\\\\\\\&=\\dfrac{x^2-6x+9}{\\color{blue}{4(x^2-x-6)}} \\cdot \\dfrac{\\color{blue}{20x^3}}{\\color{blue}{5(x^2-9)}}&&\\text{Factor}\\\\\\\\&=\\dfrac{(x-3)(x-3)}{\\color{blue}{4(x-3)(x+2)}} \\cdot \\dfrac{\\color{blue}{20x^3}}{\\color{blue}{5(x-3)(x+3)}}&&\\text{Factor. Restrictions: }x\\neq \\pm 3,\\,-2,\\,0\\\\\\\\&=\\dfrac{\\cancel{(x-3)}\\cancel{(x-3)}}{\\cancel{4}\\cancel{(x-3)}(x+2)} \\cdot \\dfrac{\\cancel{5} \\cdot \\cancel{4} \\cdot x^3}{\\cancel{5}(x+3)\\cancel{(x-3)}}&&\\text{Cancel}\\\\\\\\&=\\dfrac{x^3}{(x+2)(x+3)}&&\\text{Simplify}\\end{aligned}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Domain of [latex]\\left(\\dfrac{f}{g}\\right)(x)=\\{x\\;|\\;x\\in\\mathbb{R},\\;x\\neq\\pm 3,\\,-2,\\,0\\}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 6<\/h3>\n<p>Divide the rational functions [latex]f(x)=\\dfrac{2x^2+5x-12}{3x^2-2x}[\/latex] and\u00a0[latex]g(x)=\\dfrac{2x^2-x-3}{3x^2+x-2}[\/latex]. State the domain of the quotient function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm104\">Show Answer<\/span><\/p>\n<div id=\"qhjm104\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(\\dfrac{f}{g}\\right)(x)=\\dfrac{x+4}{x}\\;\\;\\;x\\neq 0,\\,\\frac{2}{3},\\,\\frac{3}{2},\\,-1[\/latex]<\/p>\n<p>Domain of\u00a0[latex]\\left(\\dfrac{f}{g}\\right)(x)=\\{x\\;|\\;x\\in\\mathbb{R},\\;x\\neq 0,\\,\\frac{2}{3},\\,\\frac{3}{2},\\,-1\\}[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<h2>Multiplying and Dividing Rational Functions<\/h2>\n<p>The techniques we have learned to multiply and divide rational functions can be used to extend the number of functions being multiplied or divided. Any time we need to divide, we multiply by the reciprocal.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 9<\/h3>\n<p>Simplify: [latex]\\left(\\dfrac{f\\cdot g}{h}\\right)(x)[\/latex] when [latex]f(x)=\\dfrac{x^3+2x^2}{x+1}[\/latex],\u00a0[latex]g(x)=\\dfrac{x^2-2x-3}{x}[\/latex], and [latex]h(x)=\\dfrac{x^2-3x}{x^2+7x+10}[\/latex].<\/p>\n<p>State the domain of the product\/quotient function.<\/p>\n<h4>Solution<\/h4>\n<p>Blue highlights where restrictions on the domain are found.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\left(\\dfrac{f\\cdot g}{h}\\right)(x)&=\\dfrac{x^3+2x^2}{x+1}\\cdot\\dfrac{x^2-2x-3}{x}\\div\\dfrac{x^2-3x}{x^2+7x+10}\\\\\\\\&=\\dfrac{x^3+2x^2}{\\color{blue}{x+1}}\\cdot\\dfrac{x^2-2x-3}{\\color{blue}{x}}\\cdot \\dfrac{\\color{blue}{x^2+7x+10}}{\\color{blue}{x^2-3x}}&&\\text{Write division as multiplication of the reciprocal}\\\\\\\\&=\\dfrac{x^2(x+2)}{\\color{blue}{x+1}}\\cdot\\dfrac{(x-3)(x+1)}{\\color{blue}{x}}\\cdot \\dfrac{\\color{blue}{(x+5)(x+2)}}{\\color{blue}{x(x-3)}}&&\\text{Factor. Restrictions: }x\\neq -1,\\,0,\\,3,\\,-2,\\,-5\\\\\\\\&=\\dfrac{\\cancel{x^2}(x+2)}{\\cancel{(x+1)}}\\cdot\\dfrac{\\cancel{(x-3)}\\cancel{(x+1)}}{\\cancel{x}}\\cdot \\dfrac{(x+5)(x+2)}{\\cancel{x}\\cancel{(x-3)}}&&\\text{Cancel}\\\\\\\\&=(x+5)(x+2)^2\\end{aligned}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Domain of\u00a0[latex]\\left(\\dfrac{f\\cdot g}{h}\\right)(x)=\\{x\\;|\\;x\\in\\mathbb{R},\\;x\\neq -1,\\,0,\\,-2,\\,-5\\}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 7<\/h3>\n<p>Simplify: [latex]\\left(\\dfrac{f\\cdot h}{g}\\right)(x)[\/latex] when [latex]f(x)=\\dfrac{x^3+2x^2}{x+1}[\/latex],\u00a0[latex]g(x)=\\dfrac{x^2-2x-3}{x}[\/latex], and [latex]h(x)=\\dfrac{x^2-3x}{x^2+7x+10}[\/latex].<\/p>\n<p>State the domain of the product\/quotient function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm015\">Show Answer<\/span><\/p>\n<div id=\"qhjm015\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(\\dfrac{f\\cdot h}{g}\\right)(x)=\\dfrac{x^4}{(x+1)^2(x+5)}\\;\\;x\\neq -1,\\,0,\\,-2\\,-5[\/latex]<\/p>\n<p>Domain of\u00a0[latex]\\left(\\dfrac{f\\cdot h}{g}\\right)(x)={x\\;|\\;x\\in\\mathbb{R},\\;x\\neq -1,\\,0,\\,-2\\,-5}[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\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-2831\">\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>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><li>Adaptation and Revision. <strong>Authored by<\/strong>: Hazel McKenna and Leo Chang. <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>Ex5 Multiplying Rational Functions, Ex8 Dividing Rational Functions. <strong>Authored by<\/strong>: Leo Chang. <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>Examples 1, 2 and 9. Try It: hjm104; hjm611; hjm203; hjm616; hjm775; hjm344; hjm015. <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>SImplifying Rational Expressions. <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>Multiplying and Dividing Rational 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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 15: Rational Expressions, 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\/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><\/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\":\"cc\",\"description\":\"Unit 15: Rational Expressions, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Adaptation and Revision\",\"author\":\"Hazel McKenna and Leo Chang\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Ex5 Multiplying Rational Functions, Ex8 Dividing Rational Functions\",\"author\":\"Leo Chang\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Examples 1, 2 and 9. 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