Partial Fraction Decomposition Calculator

Work out partial fraction decomposition instantly with clear inputs, formula shown and shareable results.

About Partial Fraction Decomposition Calculator

For (px + q)/((x − a)(x − b)) with distinct roots, the cover-up rule gives A = (pa + q)/(a − b) and B = (pb + q)/(b − a). As a consistency check A + B must equal p, the coefficient of x.

Formula

Cover-up rule

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

Worked example

With the default values below, Partial Fraction Decomposition Calculator returns Partial fraction form of 2.666667/(x - 1) + 0.333333/(x + 2). Change any input above and every figure here is recalculated in your browser.

Inputs used

Numerator coefficient of x: 3
Numerator constant: 5
First root a (from x − a): 1
Second root b (from x − b): -2

Results

Partial fraction form: 2.666667/(x - 1) + 0.333333/(x + 2)
Coefficient A: 2.66666667
Coefficient B: 0.33333333
A + B (must equal p): 3.00000000

Step by step

1Values used
2Cover-up rule
3Partial fraction form= 2.666667/(x - 1) + 0.333333/(x + 2)
4Coefficient A= 2.66666667
5Coefficient B= 0.33333333
6A + B (must equal p)= 3.00000000

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For (3x + 5)/((x − 1)(x + 2))?

A = (3 + 5)/3 = 8/3 and B = (−6 + 5)/(−3) = 1/3, and 8/3 + 1/3 = 3 = p as required.

What about repeated roots?

A repeated factor (x − a)² needs terms A/(x − a) + B/(x − a)², which this two-distinct-root form does not cover.