{"id":1262,"date":"2016-10-21T21:22:58","date_gmt":"2016-10-21T21:22:58","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=1262"},"modified":"2017-04-10T19:59:22","modified_gmt":"2017-04-10T19:59:22","slug":"multiplying-and-dividing-rational-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/multiplying-and-dividing-rational-expressions\/","title":{"raw":"Multiply and Divide Rational Expressions","rendered":"Multiply and Divide Rational Expressions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Simplify a rational expression<\/li>\r\n \t<li>Multiply and divide rational expressions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\"><img class=\"alignnone size-full wp-image-3370\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/13193222\/calculator.png\" alt=\"\" width=\"251\" height=\"46\" \/><\/a>\r\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 cancelling 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.\r\n<div style=\"text-align: center;\">[latex]\\frac{{x}^{2}+8x+16}{{x}^{2}+11x+28}[\/latex]<\/div>\r\nWe can factor the numerator and denominator to rewrite the expression.\r\n<div style=\"text-align: center;\">[latex]\\frac{{\\left(x+4\\right)}^{2}}{\\left(x+4\\right)\\left(x+7\\right)}[\/latex]<\/div>\r\nThen we can simplify that expression by canceling the common factor [latex]\\left(x+4\\right)[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\frac{x+4}{x+7}[\/latex]<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a rational expression, simplify it.<\/h3>\r\n<ol>\r\n \t<li>Factor the numerator and denominator.<\/li>\r\n \t<li>Cancel any common factors.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Simplifying Rational Expressions<\/h3>\r\nSimplify [latex]\\frac{{x}^{2}-9}{{x}^{2}+4x+3}[\/latex].\r\n\r\n[reveal-answer q=\"568949\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"568949\"]\r\n[latex]\\begin{array}\\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]\r\n<h4>Analysis of the Solution<\/h4>\r\nWe can cancel the common factor because any expression divided by itself is equal to 1.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<h3>Can the [latex]{x}^{2}[\/latex] term be cancelled in the above example?<\/h3>\r\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>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSimplify [latex]\\frac{x - 6}{{x}^{2}-36}[\/latex].\r\n\r\n[reveal-answer q=\"17752\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"17752\"][latex]\\frac{1}{x+6}[\/latex][\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=110917&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=110916&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Multiplying Rational Expressions<\/h2>\r\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.\r\n<div class=\"textbox\">\r\n<h3>How To: Given two rational expressions, multiply them.<\/h3>\r\n<ol>\r\n \t<li>Factor the numerator and denominator.<\/li>\r\n \t<li>Multiply the numerators.<\/li>\r\n \t<li>Multiply the denominators.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Multiplying Rational Expressions<\/h3>\r\nMultiply the rational expressions and show the product in simplest form:\r\n<div 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]<\/div>\r\n[reveal-answer q=\"820400\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"820400\"]\r\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>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nMultiply the rational expressions and show the product in simplest form:\r\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>\r\n[reveal-answer q=\"165135\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"165135\"]\u00a0[latex]\\frac{\\left(x+5\\right)\\left(x+6\\right)}{\\left(x+2\\right)\\left(x+4\\right)}[\/latex][\/hidden-answer]\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=93841&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><\/iframe>\r\n<iframe id=\"mom4\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=93844&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Dividing Rational Expressions<\/h2>\r\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.\r\n<div style=\"text-align: center;\">[latex]\\frac{1}{x}\\cdot \\frac{3}{{x}^{2}}=\\frac{3}{{x}^{3}}[\/latex]<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given two rational expressions, divide them.<\/h3>\r\n<ol>\r\n \t<li>Rewrite as the first rational expression multiplied by the reciprocal of the second.<\/li>\r\n \t<li>Factor the numerators and denominators.<\/li>\r\n \t<li>Multiply the numerators.<\/li>\r\n \t<li>Multiply the denominators.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Dividing Rational Expressions<\/h3>\r\nDivide the rational expressions and express the quotient in simplest form:\r\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>\r\n[reveal-answer q=\"266408\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"266408\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}\\text{ }\\frac{2x^{2}+x-6}{x^{2}}\\cdot\\frac{x^{2}+2x+1}{x^{2}-4} \\hfill&amp; \\text{Rewrite as the first rational expression multiplied by the reciprocal of the second rational expression.} \\\\ \\frac{\\left(2\\times3\\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]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDivide the rational expressions and express the quotient in simplest form:\r\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>\r\n[reveal-answer q=\"396693\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"396693\"][latex]1[\/latex][\/hidden-answer]\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=93845&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe>\r\n<iframe id=\"mom6\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=93847&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Simplify a rational expression<\/li>\n<li>Multiply and divide rational expressions<\/li>\n<\/ul>\n<\/div>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3370\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/13193222\/calculator.png\" alt=\"\" width=\"251\" height=\"46\" \/><\/a><br \/>\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 cancelling 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<div style=\"text-align: center;\">[latex]\\frac{{x}^{2}+8x+16}{{x}^{2}+11x+28}[\/latex]<\/div>\n<p>We can factor the numerator and denominator to rewrite the expression.<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{{\\left(x+4\\right)}^{2}}{\\left(x+4\\right)\\left(x+7\\right)}[\/latex]<\/div>\n<p>Then we can simplify that expression by canceling the common factor [latex]\\left(x+4\\right)[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{x+4}{x+7}[\/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\">Solution<\/span><\/p>\n<div id=\"q568949\" class=\"hidden-answer\" style=\"display: none\">\n[latex]\\begin{array}\\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<h3>Can the [latex]{x}^{2}[\/latex] term be cancelled in the above example?<\/h3>\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\">Solution<\/span><\/p>\n<div id=\"q17752\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{1}{x+6}[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.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:\/\/www.myopenmath.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]\\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]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q820400\">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\">Solution<\/span><\/p>\n<div id=\"q165135\" class=\"hidden-answer\" style=\"display: none\">\u00a0[latex]\\frac{\\left(x+5\\right)\\left(x+6\\right)}{\\left(x+2\\right)\\left(x+4\\right)}[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.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:\/\/www.myopenmath.com\/multiembedq.php?id=93844&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h2>Dividing Rational Expressions<\/h2>\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\">Solution<\/span><\/p>\n<div id=\"q266408\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}\\text{ }\\frac{2x^{2}+x-6}{x^{2}}\\cdot\\frac{x^{2}+2x+1}{x^{2}-4} \\hfill& \\text{Rewrite as the first rational expression multiplied by the reciprocal of the second rational expression.} \\\\ \\frac{\\left(2\\times3\\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\">Solution<\/span><\/p>\n<div id=\"q396693\" class=\"hidden-answer\" style=\"display: none\">[latex]1[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.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:\/\/www.myopenmath.com\/multiembedq.php?id=93847&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1262\">\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. <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><\/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":21,"menu_order":11,"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 al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Question ID 110917, 110916\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC- BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 93841, 93844, 93845, 93847\",\"author\":\"Michael Jenck\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License, CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"5b19b97e-b9f4-4987-9bfb-995fbbbe8e87","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1262","chapter","type-chapter","status-publish","hentry"],"part":1203,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1262","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":8,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1262\/revisions"}],"predecessor-version":[{"id":3988,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1262\/revisions\/3988"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1203"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1262\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/media?parent=1262"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1262"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1262"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/license?post=1262"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}