Partial Fraction Decomposition Calculator

Decompose (px + q) ÷ ((x − r₁)(x − r₂)) into A/(x − r₁) + B/(x − r₂).

A 1.333

B 1.667

Formula: A = (p·r₁ + q)/(r₁ − r₂), B = (p·r₂ + q)/(r₂ − r₁)

Step-by-step with your numbers:

1. Values used:

2. p (numerator x) = 3

3. q (numerator const) = 1

4. Root r₁ = 1

5. Root r₂ = -2

7. A = 1.333

8. B = 1.667

Did we solve your problem today?

Partial fractions split a rational expression into simpler pieces — essential for integration.

The math behind it

For distinct linear roots, use the cover-up method: A is the numerator evaluated at x = r₁ divided by (r₁ − r₂), and similarly for B.

(3x + 1)/((x − 1)(x + 2)) = (4/3)/(x − 1) + (5/3)/(x + 2).

What about repeated or complex roots?

Those need extra terms (e.g. A/(x−r) + B/(x−r)²); this tool handles two distinct real roots.