Break it into two simple pieces: A messy fraction like (5x − 1)/[(x + 1)(x − 2)] can be rewritten as two simple fractions:
A/(x + 1) + B/(x − 2). Your only job is to find the numbers A and B.
IB-style question — split the fraction
Express (5x − 1)/[(x + 1)(x − 2)] in partial fractions.
Step by step
- Write the split, then multiply both sides by (x + 1)(x − 2) to clear the fractions.
- Put x = 2 — this zeroes the (x − 2) bracket, so the A-term disappears and only B is left.
- Put x = −1 — this zeroes the (x + 1) bracket, leaving only A.
- Write the answer with A = 2, B = 3.
Final answer
2/(x + 1) + 3/(x − 2).
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Zero out one bracket at a time: To find each unknown fast, substitute the x that makes a bracket zero. That deletes one term and leaves a single unknown — no simultaneous equations needed.
IB-style question — find A and B
Express (x + 7)/[(x − 1)(x + 3)] in partial fractions.
Step by step
- Clear the fractions.
- Put x = 1 (zeroes the B-bracket) → only A survives.
- Put x = −3 (zeroes the A-bracket) → only B survives.
- Write the answer.
Final answer
2/(x − 1) − 1/(x + 3).