{"id":1757,"date":"2023-10-12T00:32:06","date_gmt":"2023-10-12T00:32:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introductionrational-expressions\/"},"modified":"2023-10-12T00:32:06","modified_gmt":"2023-10-12T00:32:06","slug":"introductionrational-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introductionrational-expressions\/","title":{"raw":"Rational Expressions","rendered":"Rational Expressions"},"content":{"raw":"\n\n<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Simplify rational expressions.<\/li>\n \t<li>Multiply and divide rational expressions.<\/li>\n \t<li>Find the least common denominator of two rational expressions.<\/li>\n \t<li>Add and subtract rational expressions.<\/li>\n \t<li>Simplify complex rational expressions.<\/li>\n<\/ul>\n<\/div>\nA pastry shop has fixed costs of [latex]\\$280[\/latex] per week and variable costs of [latex]\\$9[\/latex] per box of pastries. The shop\u2019s costs per week in terms of [latex]x[\/latex], the number of boxes made, is [latex]280+9x[\/latex]. We can divide the costs per week by the number of boxes made to determine the cost per box of pastries.\n<p style=\"text-align: center;\">[latex]\\frac{280+9x}{x}[\/latex]<\/p>\nNotice that the result is a polynomial expression divided by a second polynomial expression. In this section, we will explore quotients of polynomial expressions.\n<h2>Simplifying Rational Expressions<\/h2>\nThe quotient of two polynomial expressions is called a <strong>rational expression<\/strong>. We can apply the properties of fractions to rational expressions such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let\u2019s start with the rational expression shown.\n<p style=\"text-align: center;\">[latex]\\frac{{x}^{2}+8x+16}{{x}^{2}+11x+28}[\/latex]<\/p>\n[latex]\\\\[\/latex]We can factor the numerator and denominator to rewrite the expression as [latex]\\frac{{\\left(x+4\\right)}^{2}}{\\left(x+4\\right)\\left(x+7\\right)}[\/latex]\n<div><\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<div>Then we can simplify the expression by canceling the common factor [latex]\\left(x+4\\right)[\/latex] to get [latex]\\frac{x+4}{x+7}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a rational expression, simplify it<\/h3>\n<ol>\n \t<li>Factor the numerator and denominator.<\/li>\n \t<li>Cancel any common factors.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying Rational Expressions<\/h3>\nSimplify [latex]\\frac{{x}^{2}-9}{{x}^{2}+4x+3}[\/latex].\n\n[reveal-answer q=\"568949\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"568949\"]\n\n[latex]\\begin{array}{lllllllll}\\frac{\\left(x+3\\right)\\left(x - 3\\right)}{\\left(x+3\\right)\\left(x+1\\right)}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Factor the numerator and the denominator}.\\hfill \\\\ \\frac{x - 3}{x+1}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Cancel common factor }\\left(x+3\\right).\\hfill \\end{array}[\/latex]\n<h4>Analysis of the Solution<\/h4>\nWe can cancel the common factor because any expression divided by itself is equal to 1.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<strong>Can the [latex]{x}^{2}[\/latex] term be cancelled in the above example?<\/strong>\n\n<em>No. A factor is an expression that is multiplied by another expression. The [latex]{x}^{2}[\/latex] term is not a factor of the numerator or the denominator.<\/em>\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nSimplify [latex]\\frac{x - 6}{{x}^{2}-36}[\/latex].\n\n[reveal-answer q=\"17752\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"17752\"]\n\n[latex]\\frac{1}{x+6}[\/latex][\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110917&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe>\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110916&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<h2>Multiplying Rational Expressions<\/h2>\nMultiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying rational expressions. We are often able to simplify the product of rational expressions.\n<div class=\"textbox\">\n<h3>How To: Given two rational expressions, multiply them<\/h3>\n<ol>\n \t<li>Factor the numerator and denominator.<\/li>\n \t<li>Multiply the numerators.<\/li>\n \t<li>Multiply the denominators.<\/li>\n \t<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying Rational Expressions<\/h3>\nMultiply the rational expressions and show the product in simplest form:\n<div style=\"text-align: center;\">[latex]\\frac{{x}^{2}+4x-5}{3x+18}\\cdot \\frac{2x - 1}{x+5}[\/latex]<\/div>\n<div><\/div>\n[reveal-answer q=\"820400\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"820400\"]\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}\\frac{\\left(x+5\\right)\\left(x - 1\\right)}{3\\left(x+6\\right)}\\cdot \\frac{\\left(2x - 1\\right)}{\\left(x+5\\right)}\\hfill &amp; \\text{Factor the numerator and denominator}.\\hfill \\\\ \\frac{\\left(x+5\\right)\\left(x - 1\\right)\\left(2x - 1\\right)}{3\\left(x+6\\right)\\left(x+5\\right)}\\hfill &amp; \\text{Multiply numerators and denominators}.\\hfill \\\\ \\frac{\\cancel{\\left(x+5\\right)}\\left(x - 1\\right)\\left(2x - 1\\right)}{3\\left(x+6\\right)\\cancel{\\left(x+5\\right)}}\\hfill &amp; \\text{Cancel common factors to simplify}.\\hfill \\\\ \\frac{\\left(x - 1\\right)\\left(2x - 1\\right)}{3\\left(x+6\\right)}\\hfill &amp; \\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nMultiply the rational expressions and show the product in simplest form:\n<div style=\"text-align: center;\">[latex]\\frac{{x}^{2}+11x+30}{{x}^{2}+5x+6}\\cdot \\frac{{x}^{2}+7x+12}{{x}^{2}+8x+16}[\/latex]<\/div>\n[reveal-answer q=\"165135\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"165135\"]\n\n[latex]\\frac{\\left(x+5\\right)\\left(x+6\\right)}{\\left(x+2\\right)\\left(x+4\\right)}[\/latex][\/hidden-answer]\n\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93841&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><\/iframe>\n<iframe id=\"mom4\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93844&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<h3>Dividing Rational Expressions<\/h3>\nDivision of rational expressions works the same way as division of other fractions. To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second. Using this approach, we would rewrite [latex]\\frac{1}{x}\\div \\frac{{x}^{2}}{3}[\/latex] as the product [latex]\\frac{1}{x}\\cdot \\frac{3}{{x}^{2}}[\/latex]. Once the division expression has been rewritten as a multiplication expression, we can multiply as we did before.\n<div style=\"text-align: center;\">[latex]\\frac{1}{x}\\cdot \\frac{3}{{x}^{2}}=\\frac{3}{{x}^{3}}[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>How To: Given two rational expressions, divide them<\/h3>\n<ol>\n \t<li>Rewrite as the first rational expression multiplied by the reciprocal of the second.<\/li>\n \t<li>Factor the numerators and denominators.<\/li>\n \t<li>Multiply the numerators.<\/li>\n \t<li>Multiply the denominators.<\/li>\n \t<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Dividing Rational Expressions<\/h3>\nDivide the rational expressions and express the quotient in simplest form:\n<div style=\"text-align: center;\">[latex]\\frac{2{x}^{2}+x - 6}{{x}^{2}-1}\\div \\frac{{x}^{2}-4}{{x}^{2}+2x+1}[\/latex]<\/div>\n[reveal-answer q=\"266408\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"266408\"]\n\n[latex]\\begin{array}\\text{ }\\frac{2x^{2}+x-6}{x^{2}-1}\\cdot\\frac{x^{2}+2x+1}{x^{2}-4} \\hfill&amp; \\text{Rewrite as the first fraction multiplied by the reciprocal of the second fraction.} \\\\ \\frac{\\left(2x-3\\right)\\cancel{\\left(x+2\\right)}}{\\cancel{\\left(x+1\\right)}\\left(x-1\\right)}\\cdot\\frac{\\cancel{\\left(x+1\\right)}\\left(x+1\\right)}{\\cancel{\\left(x+2\\right)}\\left(x-2\\right)} \\hfill&amp; \\text{Factor and cancel common factors.} \\\\ \\frac{\\left(2x+3\\right)\\left(x+1\\right)}{\\left(x-1\\right)\\left(x-2\\right)} \\hfill&amp; \\text{Multiply numerators and denominators.} \\\\ \\frac{2x^{2}+5x+3}{x^{2}-3x+2} \\hfill&amp; \\text{Simplify.}\\end{array}[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nDivide the rational expressions and express the quotient in simplest form:\n<div style=\"text-align: center;\">[latex]\\frac{9{x}^{2}-16}{3{x}^{2}+17x - 28}\\div \\frac{3{x}^{2}-2x - 8}{{x}^{2}+5x - 14}[\/latex]<\/div>\n[reveal-answer q=\"396693\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"396693\"]\n\n[latex]1[\/latex][\/hidden-answer]\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93845&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe>\n<iframe id=\"mom6\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93847&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<h2>Adding and Subtracting Rational Expressions<\/h2>\nAdding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let\u2019s look at an example of fraction addition.\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\frac{5}{24}+\\frac{1}{40}&amp; =&amp; \\frac{25}{120}+\\frac{3}{120}\\hfill \\\\ &amp; =&amp; \\frac{28}{120}\\hfill \\\\ &amp; =&amp; \\frac{7}{30}\\hfill \\end{array}[\/latex]<\/div>\nWe have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same thing when adding or subtracting rational expressions.\n\nThe easiest common denominator to use will be the <strong>least common denominator<\/strong>&nbsp;or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, if the factored denominators were [latex]\\left(x+3\\right)\\left(x+4\\right)[\/latex] and [latex]\\left(x+4\\right)\\left(x+5\\right)[\/latex], then the LCD would be [latex]\\left(x+3\\right)\\left(x+4\\right)\\left(x+5\\right)[\/latex].\n\nOnce we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD. We would need to multiply the expression with a denominator of [latex]\\left(x+3\\right)\\left(x+4\\right)[\/latex] by [latex]\\frac{x+5}{x+5}[\/latex] and the expression with a denominator of [latex]\\left(x+4\\right)\\left(x+5\\right)[\/latex] by [latex]\\frac{x+3}{x+3}[\/latex].\n<div class=\"textbox\">\n<h3>How To: Given two rational expressions, add or subtract them<\/h3>\n<ol>\n \t<li>Factor the numerator and denominator.<\/li>\n \t<li>Find the LCD of the expressions.<\/li>\n \t<li>Multiply the expressions by a form of 1 that changes the denominators to the LCD.<\/li>\n \t<li>Add or subtract the numerators.<\/li>\n \t<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Adding Rational Expressions<\/h3>\nAdd the rational expressions:\n<div style=\"text-align: center;\">[latex]\\frac{5}{x}+\\frac{6}{y}[\/latex]<\/div>\n[reveal-answer q=\"232817\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"232817\"]\n\nFirst, we have to find the LCD. In this case, the LCD will be [latex]xy[\/latex]. We then multiply each expression by the appropriate form of 1 to obtain [latex]xy[\/latex] as the denominator for each fraction.\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\frac{5}{x}\\cdot \\frac{y}{y}+\\frac{6}{y}\\cdot \\frac{x}{x}\\\\ \\frac{5y}{xy}+\\frac{6x}{xy}\\end{array}[\/latex]<\/div>\nNow that the expressions have the same denominator, we simply add the numerators to find the sum.\n<div style=\"text-align: center;\">[latex]\\frac{6x+5y}{xy}[\/latex]<\/div>\n[\/hidden-answer]\n<h4>Analysis of the Solution<\/h4>\nMultiplying by [latex]\\frac{y}{y}[\/latex] or [latex]\\frac{x}{x}[\/latex] does not change the value of the original expression because any number divided by itself is 1, and multiplying an expression by 1 gives the original expression.\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110918&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110919&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n&nbsp;\n<div class=\"textbox exercises\">\n<h3>Example: Subtracting Rational Expressions<\/h3>\nSubtract the rational expressions:\n<div style=\"text-align: center;\">[latex]\\frac{6}{{x}^{2}+4x+4}-\\frac{2}{{x}^{2}-4}[\/latex]<\/div>\n[reveal-answer q=\"122137\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"122137\"]\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}\\frac{6}{{\\left(x+2\\right)}^{2}}-\\frac{2}{\\left(x+2\\right)\\left(x - 2\\right)}\\hfill &amp; \\text{Factor}.\\hfill \\\\ \\frac{6}{{\\left(x+2\\right)}^{2}}\\cdot \\frac{x - 2}{x - 2}-\\frac{2}{\\left(x+2\\right)\\left(x - 2\\right)}\\cdot \\frac{x+2}{x+2}\\hfill &amp; \\text{Multiply each fraction to get the LCD as the denominator}.\\hfill \\\\ \\frac{6\\left(x - 2\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}-\\frac{2\\left(x+2\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill &amp; \\text{Multiply}.\\hfill \\\\ \\frac{6x - 12-\\left(2x+4\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill &amp; \\text{Apply distributive property}.\\hfill \\\\ \\frac{4x - 16}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill &amp; \\text{Subtract}.\\hfill \\\\ \\frac{4\\left(x - 4\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<strong>Do we have to use the LCD to add or subtract rational expressions?<\/strong>\n\n<em>No. Any common denominator will work, but it is easiest to use the LCD.<\/em>\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nSubtract the rational expressions: [latex]\\frac{3}{x+5}-\\frac{1}{x - 3}[\/latex].\n\n[reveal-answer q=\"820348\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"820348\"]\n\n[latex]\\frac{2\\left(x - 7\\right)}{\\left(x+5\\right)\\left(x - 3\\right)}[\/latex][\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=39519&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"550\"><\/iframe>\n\n<\/div>\n<h3>Simplifying Complex Rational Expressions<\/h3>\nA complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression [latex]\\frac{a}{\\frac{1}{b}+c}[\/latex] can be simplified by rewriting the numerator as the fraction [latex]\\frac{a}{1}[\/latex] and combining the expressions in the denominator as [latex]\\frac{1+bc}{b}[\/latex]. We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get [latex]\\frac{a}{1}\\cdot \\frac{b}{1+bc}[\/latex] which is equal to [latex]\\frac{ab}{1+bc}[\/latex].\n<div class=\"textbox\">\n<h3>How To: Given a complex rational expression, simplify it<\/h3>\n<ol>\n \t<li>Combine the expressions in the numerator into a single rational expression by adding or subtracting.<\/li>\n \t<li>Combine the expressions in the denominator into a single rational expression by adding or subtracting.<\/li>\n \t<li>Rewrite as the numerator divided by the denominator.<\/li>\n \t<li>Rewrite as multiplication.<\/li>\n \t<li>Multiply.<\/li>\n \t<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying Complex Rational Expressions<\/h3>\nSimplify: [latex]\\frac{y+\\frac{1}{x}}{\\frac{x}{y}}[\/latex] .\n\n[reveal-answer q=\"967019\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"967019\"]\n\nBegin by combining the expressions in the numerator into one expression.\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}y\\cdot \\frac{x}{x}+\\frac{1}{x}\\hfill &amp; \\text{Multiply by }\\frac{x}{x}\\text{to get LCD as denominator}.\\hfill \\\\ \\frac{xy}{x}+\\frac{1}{x}\\hfill &amp; \\\\ \\frac{xy+1}{x}\\hfill &amp; \\text{Add numerators}.\\hfill \\end{array}[\/latex]<\/div>\nNow the numerator is a single rational expression and the denominator is a single rational expression.\n<div style=\"text-align: center;\">[latex]\\frac{\\frac{xy+1}{x}}{\\frac{x}{y}}[\/latex]<\/div>\nWe can rewrite this as division and then multiplication.\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}\\frac{xy+1}{x}\\div \\frac{x}{y}\\hfill &amp; \\\\ \\frac{xy+1}{x}\\cdot \\frac{y}{x}\\hfill &amp; \\text{Rewrite as multiplication}\\text{.}\\hfill \\\\ \\frac{y\\left(xy+1\\right)}{{x}^{2}}\\hfill &amp; \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\n<div>[\/hidden-answer]<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nSimplify: [latex]\\frac{\\frac{x}{y}-\\frac{y}{x}}{y}[\/latex]\n\n[reveal-answer q=\"40643\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"40643\"]\n\n[latex]\\frac{{x}^{2}-{y}^{2}}{x{y}^{2}}[\/latex][\/hidden-answer]\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3078&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe>\n<iframe id=\"mom6\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3080&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"350\"><\/iframe>\n<iframe id=\"mom7\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=59554&amp;theme=oea&amp;iframe_resize_id=mom7\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<strong>Can a complex rational expression always be simplified?<\/strong>\n\n<em>Yes. We can always rewrite a complex rational expression as a simplified rational expression.<\/em>\n\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>Rational expressions can be simplified by canceling common factors in the numerator and denominator.<\/li>\n \t<li>We can multiply rational expressions by multiplying the numerators and multiplying the denominators.<\/li>\n \t<li>To divide rational expressions, multiply by the reciprocal of the second expression.<\/li>\n \t<li>Adding or subtracting rational expressions requires finding a common denominator.<\/li>\n \t<li>Complex rational expressions have fractions in the numerator or the denominator. These expressions can be simplified.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n \t<dt>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n \t<dt>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n \t<dt><strong>least common denominator<\/strong><\/dt>\n \t<dd id=\"fs-id1165133085665\">the smallest multiple that two denominators have in common<\/dd>\n<\/dl>\n<\/dt>\n<\/dl>\n<\/dt>\n \t<dt>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n \t<dt>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n \t<dt><strong>rational expression<\/strong><\/dt>\n \t<dd id=\"fs-id1165137644990\">the quotient of two polynomial expressions<\/dd>\n<\/dl>\n<\/dt>\n<\/dl>\n<\/dt>\n<\/dl>\n\n","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify rational expressions.<\/li>\n<li>Multiply and divide rational expressions.<\/li>\n<li>Find the least common denominator of two rational expressions.<\/li>\n<li>Add and subtract rational expressions.<\/li>\n<li>Simplify complex rational expressions.<\/li>\n<\/ul>\n<\/div>\n<p>A pastry shop has fixed costs of [latex]\\$280[\/latex] per week and variable costs of [latex]\\$9[\/latex] per box of pastries. The shop\u2019s costs per week in terms of [latex]x[\/latex], the number of boxes made, is [latex]280+9x[\/latex]. We can divide the costs per week by the number of boxes made to determine the cost per box of pastries.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{280+9x}{x}[\/latex]<\/p>\n<p>Notice that the result is a polynomial expression divided by a second polynomial expression. In this section, we will explore quotients of polynomial expressions.<\/p>\n<h2>Simplifying Rational Expressions<\/h2>\n<p>The quotient of two polynomial expressions is called a <strong>rational expression<\/strong>. We can apply the properties of fractions to rational expressions such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let\u2019s start with the rational expression shown.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{{x}^{2}+8x+16}{{x}^{2}+11x+28}[\/latex]<\/p>\n<p>[latex]\\\\[\/latex]We can factor the numerator and denominator to rewrite the expression as [latex]\\frac{{\\left(x+4\\right)}^{2}}{\\left(x+4\\right)\\left(x+7\\right)}[\/latex]<\/p>\n<div><\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<div>Then we can simplify the expression by canceling the common factor [latex]\\left(x+4\\right)[\/latex] to get [latex]\\frac{x+4}{x+7}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a rational expression, simplify it<\/h3>\n<ol>\n<li>Factor the numerator and denominator.<\/li>\n<li>Cancel any common factors.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying Rational Expressions<\/h3>\n<p>Simplify [latex]\\frac{{x}^{2}-9}{{x}^{2}+4x+3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q568949\">Show Solution<\/span><\/p>\n<div id=\"q568949\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{array}{lllllllll}\\frac{\\left(x+3\\right)\\left(x - 3\\right)}{\\left(x+3\\right)\\left(x+1\\right)}\\hfill & \\hfill & \\hfill & \\hfill & \\text{Factor the numerator and the denominator}.\\hfill \\\\ \\frac{x - 3}{x+1}\\hfill & \\hfill & \\hfill & \\hfill & \\text{Cancel common factor }\\left(x+3\\right).\\hfill \\end{array}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We can cancel the common factor because any expression divided by itself is equal to 1.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Can the [latex]{x}^{2}[\/latex] term be cancelled in the above example?<\/strong><\/p>\n<p><em>No. A factor is an expression that is multiplied by another expression. The [latex]{x}^{2}[\/latex] term is not a factor of the numerator or the denominator.<\/em><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify [latex]\\frac{x - 6}{{x}^{2}-36}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q17752\">Show Solution<\/span><\/p>\n<div id=\"q17752\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{1}{x+6}[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110917&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110916&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h2>Multiplying Rational Expressions<\/h2>\n<p>Multiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying rational expressions. We are often able to simplify the product of rational expressions.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given two rational expressions, multiply them<\/h3>\n<ol>\n<li>Factor the numerator and denominator.<\/li>\n<li>Multiply the numerators.<\/li>\n<li>Multiply the denominators.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying Rational Expressions<\/h3>\n<p>Multiply the rational expressions and show the product in simplest form:<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{{x}^{2}+4x-5}{3x+18}\\cdot \\frac{2x - 1}{x+5}[\/latex]<\/div>\n<div><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q820400\">Show Solution<\/span><\/p>\n<div id=\"q820400\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}\\frac{\\left(x+5\\right)\\left(x - 1\\right)}{3\\left(x+6\\right)}\\cdot \\frac{\\left(2x - 1\\right)}{\\left(x+5\\right)}\\hfill & \\text{Factor the numerator and denominator}.\\hfill \\\\ \\frac{\\left(x+5\\right)\\left(x - 1\\right)\\left(2x - 1\\right)}{3\\left(x+6\\right)\\left(x+5\\right)}\\hfill & \\text{Multiply numerators and denominators}.\\hfill \\\\ \\frac{\\cancel{\\left(x+5\\right)}\\left(x - 1\\right)\\left(2x - 1\\right)}{3\\left(x+6\\right)\\cancel{\\left(x+5\\right)}}\\hfill & \\text{Cancel common factors to simplify}.\\hfill \\\\ \\frac{\\left(x - 1\\right)\\left(2x - 1\\right)}{3\\left(x+6\\right)}\\hfill & \\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Multiply the rational expressions and show the product in simplest form:<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{{x}^{2}+11x+30}{{x}^{2}+5x+6}\\cdot \\frac{{x}^{2}+7x+12}{{x}^{2}+8x+16}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q165135\">Show Solution<\/span><\/p>\n<div id=\"q165135\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{\\left(x+5\\right)\\left(x+6\\right)}{\\left(x+2\\right)\\left(x+4\\right)}[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93841&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom4\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93844&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h3>Dividing Rational Expressions<\/h3>\n<p>Division of rational expressions works the same way as division of other fractions. To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second. Using this approach, we would rewrite [latex]\\frac{1}{x}\\div \\frac{{x}^{2}}{3}[\/latex] as the product [latex]\\frac{1}{x}\\cdot \\frac{3}{{x}^{2}}[\/latex]. Once the division expression has been rewritten as a multiplication expression, we can multiply as we did before.<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{1}{x}\\cdot \\frac{3}{{x}^{2}}=\\frac{3}{{x}^{3}}[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>How To: Given two rational expressions, divide them<\/h3>\n<ol>\n<li>Rewrite as the first rational expression multiplied by the reciprocal of the second.<\/li>\n<li>Factor the numerators and denominators.<\/li>\n<li>Multiply the numerators.<\/li>\n<li>Multiply the denominators.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Dividing Rational Expressions<\/h3>\n<p>Divide the rational expressions and express the quotient in simplest form:<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{2{x}^{2}+x - 6}{{x}^{2}-1}\\div \\frac{{x}^{2}-4}{{x}^{2}+2x+1}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266408\">Show Solution<\/span><\/p>\n<div id=\"q266408\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{array}\\text{ }\\frac{2x^{2}+x-6}{x^{2}-1}\\cdot\\frac{x^{2}+2x+1}{x^{2}-4} \\hfill& \\text{Rewrite as the first fraction multiplied by the reciprocal of the second fraction.} \\\\ \\frac{\\left(2x-3\\right)\\cancel{\\left(x+2\\right)}}{\\cancel{\\left(x+1\\right)}\\left(x-1\\right)}\\cdot\\frac{\\cancel{\\left(x+1\\right)}\\left(x+1\\right)}{\\cancel{\\left(x+2\\right)}\\left(x-2\\right)} \\hfill& \\text{Factor and cancel common factors.} \\\\ \\frac{\\left(2x+3\\right)\\left(x+1\\right)}{\\left(x-1\\right)\\left(x-2\\right)} \\hfill& \\text{Multiply numerators and denominators.} \\\\ \\frac{2x^{2}+5x+3}{x^{2}-3x+2} \\hfill& \\text{Simplify.}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Divide the rational expressions and express the quotient in simplest form:<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{9{x}^{2}-16}{3{x}^{2}+17x - 28}\\div \\frac{3{x}^{2}-2x - 8}{{x}^{2}+5x - 14}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q396693\">Show Solution<\/span><\/p>\n<div id=\"q396693\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]1[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93845&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom6\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93847&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h2>Adding and Subtracting Rational Expressions<\/h2>\n<p>Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let\u2019s look at an example of fraction addition.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\frac{5}{24}+\\frac{1}{40}& =& \\frac{25}{120}+\\frac{3}{120}\\hfill \\\\ & =& \\frac{28}{120}\\hfill \\\\ & =& \\frac{7}{30}\\hfill \\end{array}[\/latex]<\/div>\n<p>We have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same thing when adding or subtracting rational expressions.<\/p>\n<p>The easiest common denominator to use will be the <strong>least common denominator<\/strong>&nbsp;or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, if the factored denominators were [latex]\\left(x+3\\right)\\left(x+4\\right)[\/latex] and [latex]\\left(x+4\\right)\\left(x+5\\right)[\/latex], then the LCD would be [latex]\\left(x+3\\right)\\left(x+4\\right)\\left(x+5\\right)[\/latex].<\/p>\n<p>Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD. We would need to multiply the expression with a denominator of [latex]\\left(x+3\\right)\\left(x+4\\right)[\/latex] by [latex]\\frac{x+5}{x+5}[\/latex] and the expression with a denominator of [latex]\\left(x+4\\right)\\left(x+5\\right)[\/latex] by [latex]\\frac{x+3}{x+3}[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>How To: Given two rational expressions, add or subtract them<\/h3>\n<ol>\n<li>Factor the numerator and denominator.<\/li>\n<li>Find the LCD of the expressions.<\/li>\n<li>Multiply the expressions by a form of 1 that changes the denominators to the LCD.<\/li>\n<li>Add or subtract the numerators.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Adding Rational Expressions<\/h3>\n<p>Add the rational expressions:<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{5}{x}+\\frac{6}{y}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q232817\">Show Solution<\/span><\/p>\n<div id=\"q232817\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we have to find the LCD. In this case, the LCD will be [latex]xy[\/latex]. We then multiply each expression by the appropriate form of 1 to obtain [latex]xy[\/latex] as the denominator for each fraction.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\frac{5}{x}\\cdot \\frac{y}{y}+\\frac{6}{y}\\cdot \\frac{x}{x}\\\\ \\frac{5y}{xy}+\\frac{6x}{xy}\\end{array}[\/latex]<\/div>\n<p>Now that the expressions have the same denominator, we simply add the numerators to find the sum.<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{6x+5y}{xy}[\/latex]<\/div>\n<\/div>\n<\/div>\n<h4>Analysis of the Solution<\/h4>\n<p>Multiplying by [latex]\\frac{y}{y}[\/latex] or [latex]\\frac{x}{x}[\/latex] does not change the value of the original expression because any number divided by itself is 1, and multiplying an expression by 1 gives the original expression.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110918&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110919&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Subtracting Rational Expressions<\/h3>\n<p>Subtract the rational expressions:<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{6}{{x}^{2}+4x+4}-\\frac{2}{{x}^{2}-4}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q122137\">Show Solution<\/span><\/p>\n<div id=\"q122137\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}\\frac{6}{{\\left(x+2\\right)}^{2}}-\\frac{2}{\\left(x+2\\right)\\left(x - 2\\right)}\\hfill & \\text{Factor}.\\hfill \\\\ \\frac{6}{{\\left(x+2\\right)}^{2}}\\cdot \\frac{x - 2}{x - 2}-\\frac{2}{\\left(x+2\\right)\\left(x - 2\\right)}\\cdot \\frac{x+2}{x+2}\\hfill & \\text{Multiply each fraction to get the LCD as the denominator}.\\hfill \\\\ \\frac{6\\left(x - 2\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}-\\frac{2\\left(x+2\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill & \\text{Multiply}.\\hfill \\\\ \\frac{6x - 12-\\left(2x+4\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill & \\text{Apply distributive property}.\\hfill \\\\ \\frac{4x - 16}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill & \\text{Subtract}.\\hfill \\\\ \\frac{4\\left(x - 4\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Do we have to use the LCD to add or subtract rational expressions?<\/strong><\/p>\n<p><em>No. Any common denominator will work, but it is easiest to use the LCD.<\/em><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Subtract the rational expressions: [latex]\\frac{3}{x+5}-\\frac{1}{x - 3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q820348\">Show Solution<\/span><\/p>\n<div id=\"q820348\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{2\\left(x - 7\\right)}{\\left(x+5\\right)\\left(x - 3\\right)}[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=39519&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"550\"><\/iframe><\/p>\n<\/div>\n<h3>Simplifying Complex Rational Expressions<\/h3>\n<p>A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression [latex]\\frac{a}{\\frac{1}{b}+c}[\/latex] can be simplified by rewriting the numerator as the fraction [latex]\\frac{a}{1}[\/latex] and combining the expressions in the denominator as [latex]\\frac{1+bc}{b}[\/latex]. We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get [latex]\\frac{a}{1}\\cdot \\frac{b}{1+bc}[\/latex] which is equal to [latex]\\frac{ab}{1+bc}[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a complex rational expression, simplify it<\/h3>\n<ol>\n<li>Combine the expressions in the numerator into a single rational expression by adding or subtracting.<\/li>\n<li>Combine the expressions in the denominator into a single rational expression by adding or subtracting.<\/li>\n<li>Rewrite as the numerator divided by the denominator.<\/li>\n<li>Rewrite as multiplication.<\/li>\n<li>Multiply.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying Complex Rational Expressions<\/h3>\n<p>Simplify: [latex]\\frac{y+\\frac{1}{x}}{\\frac{x}{y}}[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q967019\">Show Solution<\/span><\/p>\n<div id=\"q967019\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by combining the expressions in the numerator into one expression.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}y\\cdot \\frac{x}{x}+\\frac{1}{x}\\hfill & \\text{Multiply by }\\frac{x}{x}\\text{to get LCD as denominator}.\\hfill \\\\ \\frac{xy}{x}+\\frac{1}{x}\\hfill & \\\\ \\frac{xy+1}{x}\\hfill & \\text{Add numerators}.\\hfill \\end{array}[\/latex]<\/div>\n<p>Now the numerator is a single rational expression and the denominator is a single rational expression.<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{\\frac{xy+1}{x}}{\\frac{x}{y}}[\/latex]<\/div>\n<p>We can rewrite this as division and then multiplication.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}\\frac{xy+1}{x}\\div \\frac{x}{y}\\hfill & \\\\ \\frac{xy+1}{x}\\cdot \\frac{y}{x}\\hfill & \\text{Rewrite as multiplication}\\text{.}\\hfill \\\\ \\frac{y\\left(xy+1\\right)}{{x}^{2}}\\hfill & \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify: [latex]\\frac{\\frac{x}{y}-\\frac{y}{x}}{y}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q40643\">Show Solution<\/span><\/p>\n<div id=\"q40643\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{{x}^{2}-{y}^{2}}{x{y}^{2}}[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3078&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom6\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3080&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom7\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=59554&amp;theme=oea&amp;iframe_resize_id=mom7\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Can a complex rational expression always be simplified?<\/strong><\/p>\n<p><em>Yes. We can always rewrite a complex rational expression as a simplified rational expression.<\/em><\/p>\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>Rational expressions can be simplified by canceling common factors in the numerator and denominator.<\/li>\n<li>We can multiply rational expressions by multiplying the numerators and multiplying the denominators.<\/li>\n<li>To divide rational expressions, multiply by the reciprocal of the second expression.<\/li>\n<li>Adding or subtracting rational expressions requires finding a common denominator.<\/li>\n<li>Complex rational expressions have fractions in the numerator or the denominator. These expressions can be simplified.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n<dt>\n<\/dt>\n<dt>\n<\/dt>\n<dt><strong>least common denominator<\/strong><\/dt>\n<dd id=\"fs-id1165133085665\">the smallest multiple that two denominators have in common<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n<dt>\n<\/dt>\n<dt><strong>rational expression<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">the quotient of two polynomial expressions<\/dd>\n<\/dl>\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-1757\">\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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><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@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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@5.2<\/li><li>Question ID 110917, 110916, 110918, 110919. <strong>Authored 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>. <strong>License Terms<\/strong>: IMathAS Community License CC- BY + GPL<\/li><li>Question ID 93841, 93844, 93845, 93847. <strong>Authored by<\/strong>: Michael Jenck. <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>: IMathAS Community License, CC-BY + GPL<\/li><li>Question ID 39519. <strong>Authored by<\/strong>: Roy Shahbazian. <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>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 3078, 3080. <strong>Authored by<\/strong>: Tophe Anderson. <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>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 59554. <strong>Authored by<\/strong>: Gary Parker. <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>: IMathAS Community License CC- BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: OpenStax College Algebra. <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><\/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":708740,"menu_order":17,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"College Algebra\",\"author\":\"OpenStax College Algebra\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et 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