Partial fractions

Partial fractions split one complicated fraction into a sum of simpler ones. It’s the reverse of adding fractions, and it makes integration and binomial expansions much easier.

Key facts

  • (px + q)/((x − a)(x − b)) = A/(x − a) + B/(x − b)
  • Find A and B by substituting values that make brackets zero (the cover-up method).
  • A repeated factor (x − a)² needs A/(x − a) + B/(x − a)².

Worked examples

Example 1

Split (5x + 1)/((x − 1)(x + 2)) into partial fractions.

  1. 5x + 1 = A(x + 2) + B(x − 1)
  2. x = 1: 6 = 3A, so A = 2.
  3. x = −2: −9 = −3B, so B = 3.
  4. 2/(x − 1) + 3/(x + 2)

Answer: 2/(x − 1) + 3/(x + 2)