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How Do You Add Two Rational Expressions with Different Denominators?

Simplify the expression shown on the right.

Summary

  1. We are adding these two rational expressions
  2. LCD stands for least common denominator
  3. (x+3)(x-3) is the same as x2-9
  4. Now that both rational expressions are over the same denominator, we can just add the numerators together

Notes

    1. Since the two denominators are not equal, we cannot add the numerators right away
    1. LCD stands for least common denominator
    1. x2-9 factors this way because it is a difference of squares
    1. LCD stands for least common denominator
    1. LCD stands for least common denominator
    2. (x+3)(x-3) is the least common multiple of the two denominators
    1. Notice that our first denominator is already x2-9
    1. LCD stands for least common denominator
    2. The least common denominator is (x+3)(x-3) = x2-9
    1. LCD stands for least common denominator
    1. Our first rational expression is x-5 over x2-9
    1. LCD stands for least common denominator
    1. x2-9 factors into (x+3)(x-3)
    1. LCD stands for least common denominator
    2. The second denominator is just x+3, which is part of the LCD
    1. The second denominator is just x+3, which is part of the LCD
    1. LCD stands for least common denominator
    1. LCD stands for least common denominator
    2. The LCD is (x+3)(x-3), so (x+3) needs to be multiplied by (x-3)
    3. If we multiply the denominator by (x-3), then we have to multiply the numerator by (x-3) so we don't change the value of the fraction
    1. Multiplying by (x-3) over (x-3) is the same as multiplying by 1
    1. To multiply (x+12) and (x-3) together we have to use the FOIL method
    2. x•x=x2
    3. x•(-3) + x•12 = 9x
    4. -3 • 12 = 36
    1. LCD stands for least common denominator
    1. Having positive signs in front of parentheses is just like multiplying by positive 1, so nothing changes in the parentheses
    1. x + 9x = 10x
    2. -5 - 36 = -41
    1. LCD stands for least common denominator
    1. Our answer is x2+10x-41 divided by x2-9
    1. LCD stands for least common denominator