Partial Fraction Decomposition Calculator

Decompose rational functions into partial fractions with step-by-step solutions. Handle linear factors, repeated factors, and irreducible quadratics for integration and differential equations.

Example Fractions

Linear Factors

Repeated Factors

Improper Fractions

Complex Examples

Rational Function Input

Numerator P(x)

Denominator Q(x)

Function:

(2x + 1) / (x^2 - 1)

Syntax Examples:

• Powers: x^2, x^3, 2x^2

• Linear: 3x + 5, -2x + 1

• Factored: (x + 1)(x - 2)

• Constants: 7, -3, 1/2

Partial Fractions Method

Step 1: Check Proper/Improper

If degree(numerator) ≥ degree(denominator), perform polynomial division first.

Step 2: Factor Denominator

Factor completely into linear and irreducible quadratic factors.

Step 3: Set Up Template

Create partial fraction form based on factor types and multiplicities.

Step 4: Solve for Coefficients

Clear denominators and solve the resulting system of equations.

Applications

Integration:

Decompose rational functions to integrate term by term easily.

Laplace Transforms:

Find inverse transforms by decomposing complex rational functions.

Differential Equations:

Solve linear ODEs with rational function solutions.

Control Theory:

Analyze transfer functions and system responses.