Home Calculators Partial Fraction Decomposition Calculator

Partial Fraction Decomposition

Decompose rational expressions into partial fractions step by step. Covers distinct linear, repeated linear, and irreducible quadratic factors with full algebraic working.

Rational Expression Input

Enter numerator and factored denominator. Use x as the variable.

Decomposition Type
Quick Examples
Expression
N(x) / D(x)

Polynomial in x

Enter factors e.g. (x+1)(x-3)(x+2)

Denominator Factors
Numerator Coefficients

Partial Fraction Forms

Distinct Linear (ax+b)

A/(ax+b)

Repeated Linear (ax+b)²

A/(ax+b) + B/(ax+b)²

Irreducible Quadratic

(Ax+B)/(ax²+bx+c)

Method
—
decomposition type
Expression Info
Num Degree—
Den Degree—
Proper fraction?—
Number of terms—
Unknowns to find—
Constant Values

Method Rules

Proper fractiondeg(N) < deg(D)
ImproperDivide first
Repeated factor (ax+b)ⁿn terms
Quadratic ax²+bx+c(Ax+B) numerator

Common Applications

Calculus integration∫ 1/(ax+b) = ln
Laplace transformsControl systems
Differential equationsSolving ODEs
Z-transformsSignal processing