Partial Fraction Decomposition Calculator

Decompose rational functions into partial fractions with step-by-step solutions, polynomial long division, and detailed factor analysis.

Example Fractions

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

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.

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.