Partial Fraction Decomposition

P(x)/Q(x) decomposition

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About Partial Fraction Decomposition

A partial fraction decomposition calculator. Decomposes P(x)/Q(x) into simpler fractions. Handles distinct linear, repeated linear, and irreducible quadratic factors. Select from preset rational functions. Shows step-by-step coefficients. All calculations are client-side.

Partial Fraction Decomposition Features

  • Decompose
  • Coefficients
  • Step-by-step
  • Presets
  • Verify
Partial fraction decomposition: write P(x)/Q(x) as sum of simpler fractions. Linear factor (x−a): A/(x−a). Repeated (x−a)²: A/(x−a) + B/(x−a)². Quadratic (x²+bx+c): (Ax+B)/(x²+bx+c). Multiply through, equate coefficients.

How to Use

Select a rational function:

  • P(x)/Q(x): Input
  • Decomposition: Partial fractions
  • Coefficients: A, B, C...

Method

  1. Factor denominator Q(x)
  2. Set up partial fraction form
  3. Multiply both sides by Q(x)
  4. Solve for coefficients (plug in roots or equate)

Applications

  • Integration: ∫dx/(x²−1) = ½ln|x−1/x+1|
  • Inverse Laplace: F(s) → f(t)
  • Z-transform inversion

Step-by-Step Instructions

  1. 1Select rational function.
  2. 2View factored denominator.
  3. 3Get partial fractions.
  4. 4See coefficients.
  5. 5Verify by combining.

Partial Fraction Decomposition — Frequently Asked Questions

When do I need partial fractions for integration?+

Whenever integrating a proper rational function P(x)/Q(x) where Q has degree ≥ 2. After decomposition, each fraction integrates to: A/(x−a) → A·ln|x−a|, A/(x−a)² → −A/(x−a), (Ax+B)/(x²+c) → arctan and ln terms.

What if degree(P) ≥ degree(Q)?+

First perform polynomial long division: P/Q = quotient + remainder/Q. Then decompose remainder/Q. The quotient is a polynomial (easy to integrate). Only proper fractions (degree P < degree Q) can be decomposed.

How do I handle repeated factors?+

For (x−a)³: need A/(x−a) + B/(x−a)² + C/(x−a)³. Each power gets its own term. The 'cover-up' method gives the highest-power term directly; others need simultaneous equations.

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