{"id":2922,"date":"2016-07-22T16:55:46","date_gmt":"2016-07-22T16:55:46","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=2922"},"modified":"2019-08-06T19:08:04","modified_gmt":"2019-08-06T19:08:04","slug":"read-define-and-simplify-rational-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-define-and-simplify-rational-expressions\/","title":{"raw":"Define and Simplify Rational Expressions","rendered":"Define and Simplify Rational Expressions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Define and simplify rational expressions<\/li>\r\n \t<li>Identify the domain of a rational expression<\/li>\r\n<\/ul>\r\n<\/div>\r\n<strong>Rational expressions<\/strong> are fractions that have a polynomial in the numerator, denominator, or both. Although rational expressions can seem complicated because they contain variables, they can be simplified\u00a0using the techniques used to simplify expressions such as [latex]\\frac{4x^3}{12x^2}[\/latex] combined with techniques for factoring polynomials. There are a couple ways to get yourself into trouble when working with rational expressions, equations, and functions. One of them is dividing by zero, and the other is trying to divide across addition or subtraction.\r\n<h2>Determine the Domain of a Rational Expression<\/h2>\r\nOne sure way you can break math is to divide by zero. Consider the following rational expression evaluated at\u00a0[latex]x = 2[\/latex]:\r\n<p style=\"text-align: center;\">Evaluate \u00a0[latex]\\frac{x}{x-2}[\/latex] for [latex]x=2[\/latex]<\/p>\r\n<p style=\"text-align: center;\">Substitute [latex]x=2[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\frac{2}{2-2}=\\frac{2}{0}\\end{array}[\/latex]<\/p>\r\nThis means that for the expression [latex]\\frac{x}{x-2}[\/latex],\u00a0[latex]x[\/latex] cannot be\u00a0[latex]2[\/latex] because it will result in an undefined ratio. In general, finding values for a variable that will not result in division by zero is called finding the domain. Finding the domain of a rational expression or function will help you not break math.\r\n<div class=\"textbox shaded\">\r\n<h3>Domain of a rational expression or equation<\/h3>\r\n<p style=\"text-align: left;\">The domain of a rational expression or equation\u00a0is a collection of the values for the variable that will not result in an undefined mathematical operation such as division by zero. \u00a0For <em>a<\/em> = any real number, we can notate the domain in the following way:<\/p>\r\n<p style=\"text-align: center;\">\u00a0[latex]x[\/latex] is all real numbers where [latex]x\\neq{a}[\/latex]<\/p>\r\n\r\n<\/div>\r\nThe reason you cannot divide any number <i>c<\/i> by zero [latex] \\left( \\frac{c}{0}\\,\\,=\\,\\,? \\right)[\/latex] is that you would have to find a number that when you multiply it by\u00a0[latex]0[\/latex] you would get back [latex]c \\left( ?\\,\\,\\cdot \\,\\,0\\,\\,=\\,\\,c \\right)[\/latex]. There are no numbers that can do this, so we say \u201cdivision by zero is undefined\u201d. When simplifying rational expressions, you need to pay attention to what values of the variable(s) in the expression would make the denominator equal zero. These values cannot be included in the domain, so they are called excluded values. Discard them right at the start before you go any further.\r\n\r\n(Note that although the <i>denominator<\/i> cannot be equal to\u00a0[latex]0[\/latex], the <i>numerator<\/i> can\u2014this is why you only look for excluded values in the denominator of a rational expression.)\r\n\r\nFor rational expressions, the domain will exclude values where the the denominator is\u00a0[latex]0[\/latex]. The following example illustrates finding the domain of an expression. Note that this is exactly the same algebra used to find the domain of a function.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nIdentify the domain of the expression. [latex] \\frac{x+7}{{{x}^{2}}+8x-9}[\/latex]\r\n[reveal-answer q=\"318517\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"318517\"]\r\n\r\nFind any values for\u00a0[latex]x[\/latex] that would make the denominator equal to\u00a0[latex]0[\/latex] by setting the denominator equal to\u00a0[latex]0[\/latex] and solving the equation.\r\n<p style=\"text-align: center;\">[latex]x^{2}+8x-9=0[\/latex]<\/p>\r\nSolve the equation by factoring. The solutions are the values that are excluded from the domain.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}(x+9)(x-1)=0\\\\x=-9\\,\\,\\,\\text{or}\\,\\,\\,x=1\\end{array}[\/latex]<\/p>\r\nThe domain is all real numbers except [latex]\u22129[\/latex] and [latex]1[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Simplify Rational Expressions<\/h2>\r\nBefore we dive in to\u00a0simplifying rational expressions, let us review the difference between a factor, \u00a0a term, \u00a0and an expression. \u00a0This will hopefully help you avoid another way to break math\u00a0when you are simplifying rational expressions.\r\n\r\n<strong>Factors<\/strong> are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number:\u00a0[latex]2[\/latex] and\u00a0[latex]10[\/latex] are factors of\u00a0[latex]20[\/latex], as are\u00a0[latex]4, 5, 1, 20[\/latex].\r\n\r\n<strong>Terms\u00a0<\/strong>are single numbers, or variables and numbers connected by multiplication.\u00a0[latex]-4, 6x[\/latex] and [latex]x^2[\/latex] are all terms.\r\n\r\n<strong>Expressions <\/strong>are<strong>\u00a0<\/strong>groups of terms connected by addition and subtraction. [latex]2x^2-5[\/latex] is an expression.\r\n\r\nThis distinction is important when you are required to divide. Let us use an example to show why this is important.\r\n\r\nSimplify: [latex]\\dfrac{2x^2}{12x}[\/latex]\r\n\r\nThe numerator and denominator of this fraction consist of factors. To simplify it, we can divide without being impeded by addition or subtraction.\r\n\r\n[latex]\\begin{array}{cc}\\dfrac{2x^2}{12x}\\\\=\\dfrac{2\\cdot{x}\\cdot{x}}{2\\cdot3\\cdot2\\cdot{x}}\\\\=\\dfrac{\\cancel{2}\\cdot{\\cancel{x}}\\cdot{x}}{\\cancel{2}\\cdot3\\cdot2\\cdot{\\cancel{x}}}\\end{array}[\/latex]\r\n\r\nWe can do this because [latex]\\frac{2}{2}=1\\text{ and }\\frac{x}{x}=1[\/latex], so our expression simplifies to [latex]\\dfrac{x}{6}[\/latex].\r\n\r\nCompare that to\u00a0the expression [latex]\\dfrac{2x^2+x}{12-2x}[\/latex]. Notice the denominator and numerator consist of two terms connected by addition and subtraction. \u00a0We have to tip-toe around the addition and subtraction. When asked to simplify, it is tempting to want to cancel out like terms as we did when we just had factors. But you cannot do that, it will break math!\r\n\r\n[caption id=\"attachment_2989\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-2989\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/22183401\/Screen-Shot-2016-07-22-at-11.32.38-AM-300x199.png\" alt=\"Shattered pottery strewn across the floor.\" width=\"300\" height=\"199\" \/> Be careful not to break math when working with rational expressions.[\/caption]\r\n\r\nIn the examples that follow, the numerator and the denominator are polynomials with more than one term, and we will show you how to properly simplify them by factoring. This turns expressions connected by addition and subtraction into terms connected by multiplication.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify and state the domain for the expression.\u00a0[latex] \\frac{x+3}{{{x}^{2}}+12x+27}[\/latex]\r\n[reveal-answer q=\"623785\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"623785\"]\r\n\r\nTo find the domain (and the excluded values), find the values where the denominator is equal to\u00a0[latex]0[\/latex]. Factor the quadratic and apply the zero product principle.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}x+3=0\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,x+9=0\\\\x=0-3\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,x=0-9\\\\x=-3\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,x=-9\\end{array}[\/latex]<\/p>\r\nThe domain is all real numbers except [latex]x=-3[\/latex] or [latex]x=-9[\/latex].\r\n\r\nFactor the numerator and denominator. \u00a0Identify the factors that are the same in the numerator and denominator then simplify.\r\n<p style=\"text-align: center;\">[latex]\\large\\begin{array}{c}\\frac{x+3}{x^{2}+12x+27}\\\\\\\\=\\frac{x+3}{\\left(x+3\\right)\\left(x+9\\right)}\\\\\\\\\\frac{\\cancel{x+3}}{\\cancel{\\left(x+3\\right)}\\left(x+9\\right)}\\\\\\\\\\normalsize=1\\cdot\\dfrac{1}{x+9}\\end{array}[\/latex]<\/p>\r\n[latex] \\frac{x+3}{{{x}^{2}}+12x+27}=\\frac{1}{x+9}[\/latex]\r\n\r\nThe domain is all real numbers except [latex]\u22123[\/latex] and [latex]\u22129[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify and state the domain for the expression.\u00a0[latex]\\frac{x^{2}+10x+24}{x^{3}-x^{2}-20x}[\/latex]\r\n[reveal-answer q=\"861958\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"861958\"]\r\n\r\nTo find the domain, determine the values where the denominator is equal to [latex]0[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x^{3}-x^{2}-20x=0\\\\x\\left(x^{2}-x-20\\right)=0\\\\x\\left(x-5\\right)\\left(x+4\\right)=0\\end{array}[\/latex]<\/p>\r\nThe domain is all real numbers except\u00a0[latex]0, 5[\/latex], and\u00a0[latex]\u22124[\/latex].\r\n\r\nTo simplify, factor the numerator and denominator of the rational expression. Identify the factors that are the same in the numerator and denominator then simplify.\r\n<p style=\"text-align: center;\">[latex] \\large\\begin{array}{c}\\frac{x^{2}+10x+24}{x^{3}-x^{2}-20x}\\\\\\\\=\\frac{\\left(x+4\\right)\\left(x+6\\right)}{x\\left(x-5\\right)\\left(x+4\\right)}\\\\\\\\=\\frac{\\cancel{\\left(x+4\\right)}\\left(x+6\\right)}{x\\left(x-5\\right)\\cancel{\\left(x+4\\right)}}\\end{array}[\/latex]<\/p>\r\nIt is acceptable to either leave the denominator in factored form or to distribute\/multiply.\r\n\r\n[latex] \\frac{x+6}{x(x-5)}[\/latex] or [latex] \\frac{x+6}{{{x}^{2}}-5x}[\/latex]\r\n\r\nThe domain is all real numbers except\u00a0[latex]0, 5[\/latex], and [latex]\u22124[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe will show one last example of simplifying a rational expression. See if you can recognize the special product in the numerator.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]\\frac{{x}^{2}-9}{{x}^{2}+4x+3}[\/latex] and state the domain.\r\n[reveal-answer q=\"773059\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"773059\"]\r\n\r\nTo find the domain, determine the values where the denominator is equal to [latex]0[\/latex]. Be sure to factor the denominator first.\r\n\r\n[latex]\\left(x+3\\right)\\left(x+1\\right)=0[\/latex]\r\n\r\nThe domain is all real numbers except\u00a0[latex]-3[\/latex] and\u00a0[latex]\u22121[\/latex].\r\n\r\nNow factor and simplify the entire rational expression. Notice the numerator is a difference of squares.\r\n<p style=\"text-align: center;\">[latex] \\large\\begin{array}{c}\\frac{{x}^{2}-9}{{x}^{2}+4x+3}\\\\\\\\=\\frac{\\left(x+3\\right)\\left(x-3\\right)}{\\left(x+3\\right)\\left(x+1\\right)}\\\\\\\\=\\frac{\\cancel{\\left(x+3\\right)}\\left(x-3\\right)}{\\cancel{\\left(x+3\\right)}\\left(x+1\\right)}\\end{array}[\/latex]<\/p>\r\n[latex]\\frac{{x}^{2}-9}{{x}^{2}+4x+3}=\\frac{x - 3}{x+1}[\/latex]\r\n\r\nDomain: [latex]x\\ne-3,-1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we present additional examples of simplifying and finding the domain of a rational expression.\r\n\r\n[embed]https:\/\/youtu.be\/tJiz5rEktBs[\/embed]\r\n<div class=\"textbox shaded\">\r\n<h3>Steps for Simplifying a Rational Expression<\/h3>\r\nTo simplify a rational expression, follow these steps:\r\n<ul>\r\n \t<li>Determine the domain. The excluded values are those values for the variable that result in the expression having a denominator of\u00a0[latex]0[\/latex].<\/li>\r\n \t<li>Factor the numerator and denominator.<\/li>\r\n \t<li>Cancel out common factors in the numerator and denominator and simplify.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nAn additional consideration for rational expressions is to determine what values are excluded from the domain. Since division by\u00a0[latex]0[\/latex] is undefined, any values of the variable that result in a denominator of\u00a0[latex]0[\/latex] must be excluded from the domain. Excluded values must be identified in the original equation, not from its factored form.Rational expressions are fractions containing polynomials. They can be simplified much like numeric fractions. To simplify a rational expression, first determine common factors of the numerator and denominator then cancel them out.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Define and simplify rational expressions<\/li>\n<li>Identify the domain of a rational expression<\/li>\n<\/ul>\n<\/div>\n<p><strong>Rational expressions<\/strong> are fractions that have a polynomial in the numerator, denominator, or both. Although rational expressions can seem complicated because they contain variables, they can be simplified\u00a0using the techniques used to simplify expressions such as [latex]\\frac{4x^3}{12x^2}[\/latex] combined with techniques for factoring polynomials. There are a couple ways to get yourself into trouble when working with rational expressions, equations, and functions. One of them is dividing by zero, and the other is trying to divide across addition or subtraction.<\/p>\n<h2>Determine the Domain of a Rational Expression<\/h2>\n<p>One sure way you can break math is to divide by zero. Consider the following rational expression evaluated at\u00a0[latex]x = 2[\/latex]:<\/p>\n<p style=\"text-align: center;\">Evaluate \u00a0[latex]\\frac{x}{x-2}[\/latex] for [latex]x=2[\/latex]<\/p>\n<p style=\"text-align: center;\">Substitute [latex]x=2[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\frac{2}{2-2}=\\frac{2}{0}\\end{array}[\/latex]<\/p>\n<p>This means that for the expression [latex]\\frac{x}{x-2}[\/latex],\u00a0[latex]x[\/latex] cannot be\u00a0[latex]2[\/latex] because it will result in an undefined ratio. In general, finding values for a variable that will not result in division by zero is called finding the domain. Finding the domain of a rational expression or function will help you not break math.<\/p>\n<div class=\"textbox shaded\">\n<h3>Domain of a rational expression or equation<\/h3>\n<p style=\"text-align: left;\">The domain of a rational expression or equation\u00a0is a collection of the values for the variable that will not result in an undefined mathematical operation such as division by zero. \u00a0For <em>a<\/em> = any real number, we can notate the domain in the following way:<\/p>\n<p style=\"text-align: center;\">\u00a0[latex]x[\/latex] is all real numbers where [latex]x\\neq{a}[\/latex]<\/p>\n<\/div>\n<p>The reason you cannot divide any number <i>c<\/i> by zero [latex]\\left( \\frac{c}{0}\\,\\,=\\,\\,? \\right)[\/latex] is that you would have to find a number that when you multiply it by\u00a0[latex]0[\/latex] you would get back [latex]c \\left( ?\\,\\,\\cdot \\,\\,0\\,\\,=\\,\\,c \\right)[\/latex]. There are no numbers that can do this, so we say \u201cdivision by zero is undefined\u201d. When simplifying rational expressions, you need to pay attention to what values of the variable(s) in the expression would make the denominator equal zero. These values cannot be included in the domain, so they are called excluded values. Discard them right at the start before you go any further.<\/p>\n<p>(Note that although the <i>denominator<\/i> cannot be equal to\u00a0[latex]0[\/latex], the <i>numerator<\/i> can\u2014this is why you only look for excluded values in the denominator of a rational expression.)<\/p>\n<p>For rational expressions, the domain will exclude values where the the denominator is\u00a0[latex]0[\/latex]. The following example illustrates finding the domain of an expression. Note that this is exactly the same algebra used to find the domain of a function.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Identify the domain of the expression. [latex]\\frac{x+7}{{{x}^{2}}+8x-9}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q318517\">Show Solution<\/span><\/p>\n<div id=\"q318517\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find any values for\u00a0[latex]x[\/latex] that would make the denominator equal to\u00a0[latex]0[\/latex] by setting the denominator equal to\u00a0[latex]0[\/latex] and solving the equation.<\/p>\n<p style=\"text-align: center;\">[latex]x^{2}+8x-9=0[\/latex]<\/p>\n<p>Solve the equation by factoring. The solutions are the values that are excluded from the domain.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}(x+9)(x-1)=0\\\\x=-9\\,\\,\\,\\text{or}\\,\\,\\,x=1\\end{array}[\/latex]<\/p>\n<p>The domain is all real numbers except [latex]\u22129[\/latex] and [latex]1[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Simplify Rational Expressions<\/h2>\n<p>Before we dive in to\u00a0simplifying rational expressions, let us review the difference between a factor, \u00a0a term, \u00a0and an expression. \u00a0This will hopefully help you avoid another way to break math\u00a0when you are simplifying rational expressions.<\/p>\n<p><strong>Factors<\/strong> are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number:\u00a0[latex]2[\/latex] and\u00a0[latex]10[\/latex] are factors of\u00a0[latex]20[\/latex], as are\u00a0[latex]4, 5, 1, 20[\/latex].<\/p>\n<p><strong>Terms\u00a0<\/strong>are single numbers, or variables and numbers connected by multiplication.\u00a0[latex]-4, 6x[\/latex] and [latex]x^2[\/latex] are all terms.<\/p>\n<p><strong>Expressions <\/strong>are<strong>\u00a0<\/strong>groups of terms connected by addition and subtraction. [latex]2x^2-5[\/latex] is an expression.<\/p>\n<p>This distinction is important when you are required to divide. Let us use an example to show why this is important.<\/p>\n<p>Simplify: [latex]\\dfrac{2x^2}{12x}[\/latex]<\/p>\n<p>The numerator and denominator of this fraction consist of factors. To simplify it, we can divide without being impeded by addition or subtraction.<\/p>\n<p>[latex]\\begin{array}{cc}\\dfrac{2x^2}{12x}\\\\=\\dfrac{2\\cdot{x}\\cdot{x}}{2\\cdot3\\cdot2\\cdot{x}}\\\\=\\dfrac{\\cancel{2}\\cdot{\\cancel{x}}\\cdot{x}}{\\cancel{2}\\cdot3\\cdot2\\cdot{\\cancel{x}}}\\end{array}[\/latex]<\/p>\n<p>We can do this because [latex]\\frac{2}{2}=1\\text{ and }\\frac{x}{x}=1[\/latex], so our expression simplifies to [latex]\\dfrac{x}{6}[\/latex].<\/p>\n<p>Compare that to\u00a0the expression [latex]\\dfrac{2x^2+x}{12-2x}[\/latex]. Notice the denominator and numerator consist of two terms connected by addition and subtraction. \u00a0We have to tip-toe around the addition and subtraction. When asked to simplify, it is tempting to want to cancel out like terms as we did when we just had factors. But you cannot do that, it will break math!<\/p>\n<div id=\"attachment_2989\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2989\" class=\"size-medium wp-image-2989\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/22183401\/Screen-Shot-2016-07-22-at-11.32.38-AM-300x199.png\" alt=\"Shattered pottery strewn across the floor.\" width=\"300\" height=\"199\" \/><\/p>\n<p id=\"caption-attachment-2989\" class=\"wp-caption-text\">Be careful not to break math when working with rational expressions.<\/p>\n<\/div>\n<p>In the examples that follow, the numerator and the denominator are polynomials with more than one term, and we will show you how to properly simplify them by factoring. This turns expressions connected by addition and subtraction into terms connected by multiplication.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify and state the domain for the expression.\u00a0[latex]\\frac{x+3}{{{x}^{2}}+12x+27}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q623785\">Show Solution<\/span><\/p>\n<div id=\"q623785\" class=\"hidden-answer\" style=\"display: none\">\n<p>To find the domain (and the excluded values), find the values where the denominator is equal to\u00a0[latex]0[\/latex]. Factor the quadratic and apply the zero product principle.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}x+3=0\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,x+9=0\\\\x=0-3\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,x=0-9\\\\x=-3\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,x=-9\\end{array}[\/latex]<\/p>\n<p>The domain is all real numbers except [latex]x=-3[\/latex] or [latex]x=-9[\/latex].<\/p>\n<p>Factor the numerator and denominator. \u00a0Identify the factors that are the same in the numerator and denominator then simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\large\\begin{array}{c}\\frac{x+3}{x^{2}+12x+27}\\\\\\\\=\\frac{x+3}{\\left(x+3\\right)\\left(x+9\\right)}\\\\\\\\\\frac{\\cancel{x+3}}{\\cancel{\\left(x+3\\right)}\\left(x+9\\right)}\\\\\\\\\\normalsize=1\\cdot\\dfrac{1}{x+9}\\end{array}[\/latex]<\/p>\n<p>[latex]\\frac{x+3}{{{x}^{2}}+12x+27}=\\frac{1}{x+9}[\/latex]<\/p>\n<p>The domain is all real numbers except [latex]\u22123[\/latex] and [latex]\u22129[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify and state the domain for the expression.\u00a0[latex]\\frac{x^{2}+10x+24}{x^{3}-x^{2}-20x}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q861958\">Show Solution<\/span><\/p>\n<div id=\"q861958\" class=\"hidden-answer\" style=\"display: none\">\n<p>To find the domain, determine the values where the denominator is equal to [latex]0[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x^{3}-x^{2}-20x=0\\\\x\\left(x^{2}-x-20\\right)=0\\\\x\\left(x-5\\right)\\left(x+4\\right)=0\\end{array}[\/latex]<\/p>\n<p>The domain is all real numbers except\u00a0[latex]0, 5[\/latex], and\u00a0[latex]\u22124[\/latex].<\/p>\n<p>To simplify, factor the numerator and denominator of the rational expression. Identify the factors that are the same in the numerator and denominator then simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\large\\begin{array}{c}\\frac{x^{2}+10x+24}{x^{3}-x^{2}-20x}\\\\\\\\=\\frac{\\left(x+4\\right)\\left(x+6\\right)}{x\\left(x-5\\right)\\left(x+4\\right)}\\\\\\\\=\\frac{\\cancel{\\left(x+4\\right)}\\left(x+6\\right)}{x\\left(x-5\\right)\\cancel{\\left(x+4\\right)}}\\end{array}[\/latex]<\/p>\n<p>It is acceptable to either leave the denominator in factored form or to distribute\/multiply.<\/p>\n<p>[latex]\\frac{x+6}{x(x-5)}[\/latex] or [latex]\\frac{x+6}{{{x}^{2}}-5x}[\/latex]<\/p>\n<p>The domain is all real numbers except\u00a0[latex]0, 5[\/latex], and [latex]\u22124[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We will show one last example of simplifying a rational expression. See if you can recognize the special product in the numerator.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify [latex]\\frac{{x}^{2}-9}{{x}^{2}+4x+3}[\/latex] and state the domain.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q773059\">Show Solution<\/span><\/p>\n<div id=\"q773059\" class=\"hidden-answer\" style=\"display: none\">\n<p>To find the domain, determine the values where the denominator is equal to [latex]0[\/latex]. Be sure to factor the denominator first.<\/p>\n<p>[latex]\\left(x+3\\right)\\left(x+1\\right)=0[\/latex]<\/p>\n<p>The domain is all real numbers except\u00a0[latex]-3[\/latex] and\u00a0[latex]\u22121[\/latex].<\/p>\n<p>Now factor and simplify the entire rational expression. Notice the numerator is a difference of squares.<\/p>\n<p style=\"text-align: center;\">[latex]\\large\\begin{array}{c}\\frac{{x}^{2}-9}{{x}^{2}+4x+3}\\\\\\\\=\\frac{\\left(x+3\\right)\\left(x-3\\right)}{\\left(x+3\\right)\\left(x+1\\right)}\\\\\\\\=\\frac{\\cancel{\\left(x+3\\right)}\\left(x-3\\right)}{\\cancel{\\left(x+3\\right)}\\left(x+1\\right)}\\end{array}[\/latex]<\/p>\n<p>[latex]\\frac{{x}^{2}-9}{{x}^{2}+4x+3}=\\frac{x - 3}{x+1}[\/latex]<\/p>\n<p>Domain: [latex]x\\ne-3,-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we present additional examples of simplifying and finding the domain of a rational expression.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify and Give the Domain of Rational Expressions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/tJiz5rEktBs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox shaded\">\n<h3>Steps for Simplifying a Rational Expression<\/h3>\n<p>To simplify a rational expression, follow these steps:<\/p>\n<ul>\n<li>Determine the domain. The excluded values are those values for the variable that result in the expression having a denominator of\u00a0[latex]0[\/latex].<\/li>\n<li>Factor the numerator and denominator.<\/li>\n<li>Cancel out common factors in the numerator and denominator and simplify.<\/li>\n<\/ul>\n<\/div>\n<h2>Summary<\/h2>\n<p>An additional consideration for rational expressions is to determine what values are excluded from the domain. Since division by\u00a0[latex]0[\/latex] is undefined, any values of the variable that result in a denominator of\u00a0[latex]0[\/latex] must be excluded from the domain. Excluded values must be identified in the original equation, not from its factored form.Rational expressions are fractions containing polynomials. They can be simplified much like numeric fractions. To simplify a rational expression, first determine common factors of the numerator and denominator then cancel them out.<\/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-2922\">\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>Screenshot: Breaking Math. <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>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>Simplify and Give the Domain of Rational Expressions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/tJiz5rEktBs\">https:\/\/youtu.be\/tJiz5rEktBs<\/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 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><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at : http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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":21,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Screenshot: Breaking Math\",\"author\":\"\",\"organization\":\"Lumen 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