{"id":896,"date":"2024-04-30T20:05:53","date_gmt":"2024-04-30T20:05:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=896"},"modified":"2024-11-20T02:44:04","modified_gmt":"2024-11-20T02:44:04","slug":"rational-expressions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/rational-expressions-learn-it-2\/","title":{"raw":"Rational Expressions: Learn It 2","rendered":"Rational Expressions: Learn It 2"},"content":{"raw":"

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\r\n
How To: Given two rational expressions, multiply them<\/strong>\r\n
    \r\n \t
  1. Factor the numerator and denominator.<\/li>\r\n \t
  2. Multiply the numerators.<\/li>\r\n \t
  3. Multiply the denominators.<\/li>\r\n \t
  4. Simplify.<\/li>\r\n<\/ol>\r\n<\/section>
    Multiply the rational expressions and show the product in simplest form:\r\n
    [latex]\\dfrac{{x}^{2}+4x-5}{3x+18}\\cdot \\dfrac{2x - 1}{x+5}[\/latex]<\/div>\r\n
    <\/div>\r\n[reveal-answer q=\"820400\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"820400\"]\r\n

    [latex]\\begin{array}{cc}\\dfrac{\\left(x+5\\right)\\left(x - 1\\right)}{3\\left(x+6\\right)}\\cdot \\dfrac{\\left(2x - 1\\right)}{\\left(x+5\\right)}\\hfill & \\text{Factor the numerator and denominator}.\\hfill \\\\ \\dfrac{\\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 \\\\ \\dfrac{\\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 \\\\ \\dfrac{\\left(x - 1\\right)\\left(2x - 1\\right)}{3\\left(x+6\\right)}\\hfill & \\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm2_question hide_question_numbers=1]18899[\/ohm2_question]<\/section>\r\n

    Dividing Rational Expressions<\/h2>\r\nDivision of rational expressions works the same way as division of other fractions.\r\n\r\n
    To divide one fraction by another, you multiply the first fraction by the reciprocal (inverse) of the second. This method simplifies the process and avoids direct division:\r\n

    [latex]\\dfrac{a}{b}\\div\\dfrac{c}{d}=\\dfrac{a}{b}\\cdot\\dfrac{d}{c}=\\dfrac{ad}{bc}[\/latex]<\/p>\r\n\r\n<\/section>

    How To: Given two rational expressions, divide them<\/strong>\r\n
      \r\n \t
    1. Rewrite as the first rational expression multiplied by the reciprocal of the second.<\/li>\r\n \t
    2. Factor the numerators and denominators.<\/li>\r\n \t
    3. Multiply the numerators.<\/li>\r\n \t
    4. Multiply the denominators.<\/li>\r\n \t
    5. Simplify.<\/li>\r\n<\/ol>\r\n<\/section>
      Divide the rational expressions and express the quotient in simplest form:\r\n
      [latex]\\dfrac{2{x}^{2}+x - 6}{{x}^{2}-1}\\div \\dfrac{{x}^{2}-4}{{x}^{2}+2x+1}[\/latex]<\/div>\r\n[reveal-answer q=\"266408\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"266408\"]\r\n\r\n[latex]\\begin{array}\\text{ }\\dfrac{2x^{2}+x-6}{x^{2}-1}\\cdot\\dfrac{x^{2}+2x+1}{x^{2}-4} \\hfill& \\text{Rewrite as the first fraction multiplied by the reciprocal of the second fraction.} \\\\ \\dfrac{\\left(2x-3\\right)\\cancel{\\left(x+2\\right)}}{\\cancel{\\left(x+1\\right)}\\left(x-1\\right)}\\cdot\\dfrac{\\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.} \\\\ \\dfrac{\\left(2x-3\\right)\\left(x+1\\right)}{\\left(x-1\\right)\\left(x-2\\right)} \\hfill& \\text{Multiply numerators and denominators.} \\\\ \\dfrac{2x^{2}-x-3}{x^{2}-3x+2} \\hfill& \\text{Simplify.}\\end{array}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
      [ohm2_question hide_question_numbers=1]18901[\/ohm2_question]<\/section>","rendered":"

      Multiplying Rational Expressions<\/h2>\n

      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

      How To: Given two rational expressions, multiply them<\/strong><\/p>\n
        \n
      1. Factor the numerator and denominator.<\/li>\n
      2. Multiply the numerators.<\/li>\n
      3. Multiply the denominators.<\/li>\n
      4. Simplify.<\/li>\n<\/ol>\n<\/section>\n
        Multiply the rational expressions and show the product in simplest form:<\/p>\n
        [latex]\\dfrac{{x}^{2}+4x-5}{3x+18}\\cdot \\dfrac{2x - 1}{x+5}[\/latex]<\/div>\n
        <\/div>\n