Partial Fraction Decomposition Calculator
Split a proper rational function with two distinct linear denominator factors into a sum of simpler fractions using the cover-up method.
- 1
Numerator for A
3 × 1 + 5 = 8 - 2
Denominator (r1 − r2)
1 − -2 = 3 - 3
Coefficient A
8 ÷ 3 = 2.6667 - 4
Numerator for B
3 × -2 + 5 = -1 - 5
Coefficient B
-1 ÷ (-2 − 1) = 0.3333Cover-up method: substitute x = r2 into the numerator and divide by (r2 − r1).
How does this calculator work?
For (px+q)/((x−r1)(x−r2)) with distinct roots, the cover-up method gives A = (p·r1+q)/(r1−r2) and B = (p·r2+q)/(r2−r1). The expression then equals A/(x−r1) + B/(x−r2). Enter p, q, r1, and r2 to see both coefficients and the full decomposed form instantly.
Formula
How this is calculated
This calculator decomposes a proper rational expression of the form (px + q) / ((x − r1)(x − r2)) into the sum A/(x − r1) + B/(x − r2). You supply the numerator coefficients p and q (where the numerator is the linear polynomial px + q) and the two distinct denominator roots r1 and r2, so the denominator factors are (x − r1) and (x − r2).
The coefficients are found with the cover-up (Heaviside) method. To get A, multiply both sides by (x − r1) and substitute x = r1, which cancels that factor and leaves A = (p·r1 + q) / (r1 − r2). Symmetrically, B = (p·r2 + q) / (r2 − r1). This avoids solving a system of equations and works directly because each substitution isolates one unknown.
The roots r1 and r2 must be distinct; if r1 = r2 the denominator is (x − r1)² (a repeated factor) and a different form A/(x − r1) + B/(x − r1)² is required, so the calculator returns a warning. The numerator degree must be less than the denominator degree (here at most 1 < 2) for the decomposition to be proper, which the px + q form guarantees.
Frequently asked questions
It is a shortcut for finding partial-fraction coefficients: multiply through by a single linear factor and substitute that factor's root. The factor "covers up" and you read off the coefficient directly, here A = (p·r1 + q)/(r1 − r2).
If r1 = r2 the denominator has a repeated factor (x − r1)², and the correct decomposition uses A/(x − r1) + B/(x − r1)². The simple two-term form only applies to distinct linear factors.
No. The fraction must be proper (numerator degree less than denominator degree). The px + q numerator keeps degree 1 below the degree-2 denominator. For higher-degree numerators, first do polynomial long division.
Also known as
TG we-Calculate Editorial Team. (2026). Partial Fraction Decomposition Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/partial-fraction-decomposition-calculator
TG we-Calculate Editorial Team. "Partial Fraction Decomposition Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/partial-fraction-decomposition-calculator.
TG we-Calculate Editorial Team, "Partial Fraction Decomposition Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/partial-fraction-decomposition-calculator
@misc{wecalculate_partial_fraction_decomposition_calculator, title = {Partial Fraction Decomposition Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/partial-fraction-decomposition-calculator}}, year = {2026}, note = {TG we-Calculate} }
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