Partial fraction decomposition
Decompose any rational expression into partial fractions. Shows complete step-by-step work including finding coefficients.
What is partial fraction decomposition?
Partial fraction decomposition breaks a complex rational expression (fraction with polynomials) into a sum of simpler fractions. For example:
(2x+3)/(x²−1) = A/(x−1) + B/(x+1)
This technique is essential for integrating rational functions in calculus.
When to use partial fractions
Integration: ∫(2x+3)/(x²−1) dx is hard directly, but ∫[A/(x−1) + B/(x+1)] dx is easy — it's just logarithms.
Inverse Laplace transforms: Engineering and control theory use partial fractions to break transfer functions into simpler components.
Simplification: Sometimes the decomposed form reveals structure that the original fraction hides.
Types of factors
Distinct linear: (x−a) → A/(x−a)
Repeated linear: (x−a)² → A/(x−a) + B/(x−a)²
Irreducible quadratic: (x²+bx+c) → (Ax+B)/(x²+bx+c)
Step-by-step method
1. Factor the denominator completely.
2. Write a partial fraction template with unknown coefficients A, B, C, etc.
3. Multiply both sides by the denominator to clear fractions.
4. Solve for coefficients by substituting convenient values of x (roots of factors) or by comparing coefficients.
Frequently asked questions
What if the degree of the numerator is ≥ the denominator?
Perform polynomial long division first, then decompose the remainder. This calculator handles this automatically.
Can I decompose expressions with complex roots?
Yes — irreducible quadratic factors (no real roots) get the form (Ax+B)/(x²+bx+c) in the decomposition.