# Partial Fraction Decomposition Calculator

URL: https://speedycalculator.com/calculators/math/partial-fraction-decomposition/
Last updated: 2026-10-30

> The partial fraction decomposition calculator splits a rational function with distinct linear factors (or a repeated linear factor) into simpler fractions using the cover-up method.

## Inputs

- Denominator
- Numerator coefficient p (of x² for 3 factors, else of x)
- Numerator coefficient q
- Numerator constant r (3 factors only)
- a
- b
- c

## Results

- Partial fractions
- Steps

## Formula

```
(px + q)/((x − a)(x − b)) = A/(x − a) + B/(x − b)
A = (pa + q)/(a − b),  B = (pb + q)/(b − a)
```

## Explanation

Partial fractions undo the addition of fractions. They make rational functions easy to integrate, to take inverse Laplace transforms of and to sum as series. Each linear factor of the denominator contributes one simple fraction.

### How to use the Partial Fraction Decomposition Calculator

- **Denominator**
- **Numerator coefficient p (of x² for 3 factors, else of x)**
- **Numerator coefficient q**
- **Numerator constant r (3 factors only)**
- **a**
- **b**
- **c**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

For distinct linear factors the coefficient of 1/(x − a) is the rest of the fraction evaluated at x = a (Heaviside's cover-up rule). For a repeated factor (x − a)², the numerator is rewritten as p(x − a) + (pa + q).

## Worked example

Inputs — Denominator: (x − a)(x − b); Numerator coefficient p (of x² for 3 factors, else of x): 3; Numerator coefficient q: 1; a: 1; b: -2.

Results — **Partial fractions = (4/3)/(x − 1) + (5/3)/(x + 2)**; Steps = (3x + 1) / ((x − 1)(x + 2)) = A/(x − 1) + B/(x + 2)
Cover-up: A = (3·1 + 1) / (1 − (−2)) = (4/3)
B = (3·(−2) + 1) / (−2 − 1) = (5/3).

## Key terms

- **Rational function**: A ratio of two polynomials.
- **Cover-up rule**: Heaviside's shortcut for coefficients of distinct linear factors.

## FAQ

**When can I use partial fractions?**
When the degree of the numerator is less than the degree of the denominator. Otherwise divide first.

**What is the cover-up method?**
To find A for (x − a), cover that factor and evaluate the rest of the fraction at x = a.

**How do repeated factors work?**
(x − a)² needs two terms: A/(x − a) + B/(x − a)².

**What about irreducible quadratics?**
They need a numerator of the form Bx + C; this calculator covers linear factors.


## Sources

- [OpenStax – Calculus Volume 2](https://openstax.org/details/books/calculus-volume-2)
- [Wolfram MathWorld – Partial fraction decomposition](https://mathworld.wolfram.com/PartialFractionDecomposition.html)