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Adding and subtracting rational expressions by factoring worksheets Grades 9–10

Factor every denominator, find the LCD, add or subtract, then simplify.

The front of the free adding and subtracting rational expressions by factoring worksheet
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Free · Worked example · 12 problems · Answer key · US Letter

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How to do it

Factor every denominator, find the LCD, then add or subtract. Cancel any factor the answer shares with the denominator.

6x−2+5x2+5x−14=6x−2+5(x−2)(x+7)\dfrac{6}{x - 2} + \dfrac{5}{x^{2} + 5x - 14} = \dfrac{6}{x - 2} + \dfrac{5}{(x - 2)(x + 7)}factor: x² + 5x − 14 = (x − 2)(x + 7)
6x−2+5(x−2)(x+7)=6(x+7)(x−2)(x+7)+5(x−2)(x+7)\dfrac{6}{x - 2} + \dfrac{5}{(x - 2)(x + 7)} = \dfrac{6(x + 7)}{(x - 2)(x + 7)} + \dfrac{5}{(x - 2)(x + 7)}rewrite each over the LCD, (x − 2)(x + 7)
6(x+7)+5(x−2)(x+7)=6x+47(x−2)(x+7)\dfrac{6(x + 7) + 5}{(x - 2)(x + 7)} = \dfrac{6x + 47}{(x - 2)(x + 7)}add the numerators, then combine like terms

Example problems

  1. 1x+7+10x2+4x−21=\dfrac{1}{x + 7} + \dfrac{10}{x^{2} + 4x - 21} =
  2. 4x+4+2x2−2x−24=\dfrac{4}{x + 4} + \dfrac{2}{x^{2} - 2x - 24} =
  3. 8x2−x−6−2x−3=\dfrac{8}{x^{2} - x - 6} - \dfrac{2}{x - 3} =
  4. 3x−3+8x2−x−6=\dfrac{3}{x - 3} + \dfrac{8}{x^{2} - x - 6} =

Rational expressions and equations

  1. Adding and subtracting rational expressions with like denominators
  2. Least common denominator of rational expressions
  3. Adding rational expressions with unlike denominators
  4. Subtracting rational expressions with unlike denominators
  5. Adding and subtracting rational expressions by factoring
  6. Simplifying complex fractions
  7. Solving rational proportions
  8. Solving rational equations