Partial fractions · Grade 12

In Grade 12 you learn: partial fractions.

Partial fractions split one fraction into a sum of simpler ones, for example (5x − 1)/((x + 1)(x − 2)) = 2/(x + 1) + 3/(x − 2). It is adding fractions in reverse, and it makes integrating and expanding much easier.

Key facts

• Distinct linear factors: (px + q)/((x − a)(x − b)) ≡ A/(x − a) + B/(x − b).
• Multiply through by the denominator, then substitute x = a and x = b to find A and B (the cover-up method).
• A repeated factor (x − a)² needs A/(x − a) + B/((x − a)²).
• If the top has the same or higher degree than the bottom (improper), divide first.