{"id":909,"date":"2024-04-30T20:54:09","date_gmt":"2024-04-30T20:54:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=909"},"modified":"2024-11-20T02:44:08","modified_gmt":"2024-11-20T02:44:08","slug":"rational-expressions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/rational-expressions-learn-it-3\/","title":{"raw":"Rational Expressions: Learn It 3","rendered":"Rational Expressions: Learn It 3"},"content":{"raw":"

Adding and Subtracting Rational Expressions<\/h2>\r\nAdding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add or subtract fractions, we need to find a common denominator.\r\n\r\nLet\u2019s look at an example of fraction addition.\r\n
[latex]\\begin{array}{ccc}\\hfill \\dfrac{5}{24}+\\dfrac{1}{40}& =& \\dfrac{25}{120}+\\dfrac{3}{120}\\hfill \\\\ & =& \\dfrac{28}{120}\\hfill \\\\ & =& \\dfrac{7}{30}\\hfill \\end{array}[\/latex]<\/div>\r\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.\r\n\r\n
How did we know what number to use for the denominator?<\/strong>In the example above, we rewrote the fractions as equivalent fractions with a common denominator of 120. Recall that we use the least common multiple of the original denominators.To find the LCM of 24 and 40, rewrite 24 and 40 as products of primes, then select the largest set of each prime appearing.[latex]24 = 2^3\\cdot3[\/latex][latex]40=2^3\\cdot5[\/latex]\r\n\r\nWe choose [latex]2^3\\cdot3\\cdot5=120[\/latex] as the LCM, since that's the largest number of factors of 2, 3, and 5 we see. The LCM is 120.\r\n\r\nWe multiply each numerator with just enough of the LCM to make each denominator 120 to get the equivalent fractions.\r\n\r\nWhen referring to fractions, we call the LCM the least common denominator<\/strong>, or the LCD.\u00a0<\/span>\r\n\r\n<\/section>
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].<\/section>
How To: Given two rational expressions, add or subtract them<\/strong>\r\n
    \r\n \t
  1. Factor the numerator and denominator.<\/li>\r\n \t
  2. Find the LCD of the expressions.<\/li>\r\n \t
  3. Multiply the expressions by a form of 1 that changes the denominators to the LCD.<\/li>\r\n \t
  4. Add or subtract the numerators.<\/li>\r\n \t
  5. Simplify.<\/li>\r\n<\/ol>\r\n<\/section>
    Although any common denominator will work for adding or subtracting rational expressions, using the least common denominator (LCD) is typically the easiest method.<\/section>
    Add the rational expressions:\r\n
    [latex]\\dfrac{5}{x}+\\dfrac{6}{y}[\/latex]<\/div>\r\n[reveal-answer q=\"232817\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"232817\"]\r\n\r\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.\r\n
    [latex]\\begin{array}{l}\\dfrac{5}{x}\\cdot \\dfrac{y}{y}+\\dfrac{6}{y}\\cdot \\dfrac{x}{x}\\\\ \\dfrac{5y}{xy}+\\dfrac{6x}{xy}\\end{array}[\/latex]<\/div>\r\nNow that the expressions have the same denominator, we simply add the numerators to find the sum.\r\n
    [latex]\\dfrac{6x+5y}{xy}[\/latex]<\/div>\r\n \r\n\r\nAnalysis of the Solution<\/strong>\r\n\r\nMultiplying by [latex]\\dfrac{y}{y}[\/latex] or [latex]\\dfrac{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.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
    Subtract the rational expressions:\r\n
    [latex]\\dfrac{6}{{x}^{2}+4x+4}-\\dfrac{2}{{x}^{2}-4}[\/latex]<\/div>\r\n[reveal-answer q=\"122137\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"122137\"]\r\n

    [latex]\\begin{array}{cc}\\dfrac{6}{{\\left(x+2\\right)}^{2}}-\\dfrac{2}{\\left(x+2\\right)\\left(x - 2\\right)}\\hfill & \\text{Factor}.\\hfill \\\\ \\dfrac{6}{{\\left(x+2\\right)}^{2}}\\cdot \\dfrac{x - 2}{x - 2}-\\dfrac{2}{\\left(x+2\\right)\\left(x - 2\\right)}\\cdot \\dfrac{x+2}{x+2}\\hfill & \\text{Multiply each fraction to get the LCD as the denominator}.\\hfill \\\\ \\dfrac{6\\left(x - 2\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}-\\dfrac{2\\left(x+2\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill & \\text{Multiply}.\\hfill \\\\ \\dfrac{6x - 12-\\left(2x+4\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill & \\text{Apply distributive property}.\\hfill \\\\ \\dfrac{4x - 16}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill & \\text{Subtract}.\\hfill \\\\ \\dfrac{4\\left(x - 4\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm2_question hide_question_numbers=1]18904[\/ohm2_question]<\/section>
    [ohm2_question hide_question_numbers=1]18905[\/ohm2_question]<\/section>","rendered":"

    Adding and Subtracting Rational Expressions<\/h2>\n

    Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add or subtract fractions, we need to find a common denominator.<\/p>\n

    Let\u2019s look at an example of fraction addition.<\/p>\n

    [latex]\\begin{array}{ccc}\\hfill \\dfrac{5}{24}+\\dfrac{1}{40}& =& \\dfrac{25}{120}+\\dfrac{3}{120}\\hfill \\\\ & =& \\dfrac{28}{120}\\hfill \\\\ & =& \\dfrac{7}{30}\\hfill \\end{array}[\/latex]<\/div>\n

    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

    How did we know what number to use for the denominator?<\/strong>In the example above, we rewrote the fractions as equivalent fractions with a common denominator of 120. Recall that we use the least common multiple of the original denominators.To find the LCM of 24 and 40, rewrite 24 and 40 as products of primes, then select the largest set of each prime appearing.[latex]24 = 2^3\\cdot3[\/latex][latex]40=2^3\\cdot5[\/latex]<\/p>\n

    We choose [latex]2^3\\cdot3\\cdot5=120[\/latex] as the LCM, since that’s the largest number of factors of 2, 3, and 5 we see. The LCM is 120.<\/p>\n

    We multiply each numerator with just enough of the LCM to make each denominator 120 to get the equivalent fractions.<\/p>\n

    When referring to fractions, we call the LCM the least common denominator<\/strong>, or the LCD.\u00a0<\/span><\/p>\n<\/section>\n

    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].<\/section>\n
    How To: Given two rational expressions, add or subtract them<\/strong><\/p>\n
      \n
    1. Factor the numerator and denominator.<\/li>\n
    2. Find the LCD of the expressions.<\/li>\n
    3. Multiply the expressions by a form of 1 that changes the denominators to the LCD.<\/li>\n
    4. Add or subtract the numerators.<\/li>\n
    5. Simplify.<\/li>\n<\/ol>\n<\/section>\n
      Although any common denominator will work for adding or subtracting rational expressions, using the least common denominator (LCD) is typically the easiest method.<\/section>\n
      Add the rational expressions:<\/p>\n
      [latex]\\dfrac{5}{x}+\\dfrac{6}{y}[\/latex]<\/div>\n