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Calculator Answers

Partial Fraction Decomposition Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A partial fraction decomposition calculator is an advanced algebra and calculus utility that breaks down a complex rational function into a sum of simpler rational expressions. The calculator factors the denominator polynomial, determines the appropriate form of the decomposition coefficients based on linear or quadratic factors, and solves the resulting system of linear equations to find the constant numerators. It provides the expanded mathematical steps required to solve for each coefficient. Calculus students use this tool to simplify rational integrands for integration.

Enter the numerator (ax + b) and the two distinct real roots of the denominator. The calculator returns A and B in the decomposition A / (x − r₁) + B / (x − r₂).

Quick Answer

Decompose complex rational expressions into simpler partial fractions. Enter the numerator and denominator polynomials to see the step-by-step algebraic expansion.

Numerator and roots

The form supported is (ax + b) / ((x − r₁)(x − r₂)), where r₁ and r₂ are distinct real roots of the denominator.

Numerator coefficients

Denominator roots

Scope

Supported: distinct linear factors in the denominator. The numerator is a degree-1 polynomial.

Out of scope: repeated factors (A/(x−r) + B/(x−r)²), irreducible quadratic factors (Ax + B over x² + 1), and numerators of higher degree than the denominator (use polynomial long division first).

Partial fractions

Decomposition

-4 / (x − 1) + 7 / (x − 2)

Original: (3x + 1) / ((x − 1)(x − 2))

A-4
B7
StepA = (a·r₁ + b)/(r₁ − r₂) = (3·1 + 1)/(1 − 2) = -4; B = (a·r₂ + b)/(r₂ − r₁) = (3·2 + 1)/(2 − 1) = 7

Partial fraction decomposition is the standard preparation step for integrating rational functions and for inverse Laplace transforms.

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Examples

(3x + 1) / ((x − 1)(x − 2))

A = −4, B = 7

(x) / ((x − 1)(x + 1))

A = 0.5, B = 0.5

(2x + 5) / ((x − 0)(x − 3))

A = −1.667, B = 3.667

(1) / ((x − 1)(x − 2))

A = −1, B = 1

How it works

Multiply both sides of (ax + b) / ((x − r₁)(x − r₂)) = A/(x − r₁) + B/(x − r₂) by the common denominator and pick clever x values to isolate A and B.

A · A = (a·r₁ + b) / (r₁ − r₂)

B · B = (a·r₂ + b) / (r₂ − r₁)

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Frequently asked questions

A way to rewrite a rational function as a sum of simpler rational functions. For example, (3x + 1) / ((x − 1)(x − 2)) can be split into A / (x − 1) + B / (x − 2). The simpler pieces are easier to integrate or invert.

Multiply both sides by the common denominator, then evaluate at strategic x values (the roots of the original denominator). The calculator uses the closed-form A = (a·r₁ + b)/(r₁ − r₂) and B = (a·r₂ + b)/(r₂ − r₁).

When the denominator has a repeated factor (x − r)², the decomposition takes a different shape: A/(x − r) + B/(x − r)². That case is out of scope for this calculator.

If the denominator includes a factor like x² + 1 that has no real roots, the partial fraction piece uses (Ax + B)/(x² + 1). That is also out of scope for this calculator's distinct-linear-factor form.

Use polynomial long division first to write the rational function as a polynomial plus a proper rational remainder. Then decompose the remainder. The polynomial long division calculator handles the first step.

Mostly in integration (so you can integrate each simpler fraction with the log rule) and in inverse Laplace transforms (so you can invert each piece using a standard table).