Partial Fraction Decomposition — Definition, Formula & Examples

Partial fraction decomposition is a technique that rewrites a single complicated fraction (with a polynomial denominator) as a sum of simpler fractions. Each simpler fraction has a denominator that is a factor of the original denominator.

Given a proper rational expression $\frac{P(x)}{Q(x)}$ where $\deg(P) < \deg(Q)$ and $Q(x)$ factors into linear or irreducible quadratic factors, partial fraction decomposition expresses $\frac{P(x)}{Q(x)}$ as a sum of fractions whose denominators are powers of those irreducible factors and whose numerators are polynomials of degree less than the corresponding factor.

Key Formula

\frac{P(x)}{(x - a)(x - b)} = \frac{A}{x - a} + \frac{B}{x - b}

Where:

$P(x)$ = Polynomial numerator with degree less than the denominator's degree
$a, b$ = Roots of the denominator (distinct linear factors)
$A, B$ = Unknown constants to be determined

How It Works

First, confirm the rational expression is proper; if not, perform polynomial long division. Next, factor the denominator completely. Set up a template with unknown constants: each distinct linear factor

(ax + b)

contributes a term

\frac{A}{ax + b}

, and each irreducible quadratic factor contributes

\frac{Ax + B}{ax^2 + bx + c}

. Repeated factors get multiple terms with increasing powers. Multiply both sides by the original denominator to clear fractions, then solve for the unknown constants by substituting strategic values of

x

or by equating coefficients of like powers.

Worked Example

Problem: Decompose into partial fractions:

\frac{5x + 1}{(x + 1)(x - 2)}

Set up the template: Write the expression as a sum of fractions with unknown constants.

\frac{5x + 1}{(x + 1)(x - 2)} = \frac{A}{x + 1} + \frac{B}{x - 2}

Clear the denominators: Multiply both sides by

(x + 1)(x - 2)

5x + 1 = A(x - 2) + B(x + 1)

Solve for A: Substitute

x = -1

to eliminate

B

5(-1) + 1 = A(-1 - 2) + 0 \implies -4 = -3A \implies A = \frac{4}{3}

Solve for B: Substitute

x = 2

to eliminate

A

5(2) + 1 = 0 + B(2 + 1) \implies 11 = 3B \implies B = \frac{11}{3}

Write the result: Replace the constants in the template.

\frac{5x + 1}{(x + 1)(x - 2)} = \frac{4/3}{x + 1} + \frac{11/3}{x - 2}

Answer:

\dfrac{5x + 1}{(x + 1)(x - 2)} = \dfrac{4}{3(x + 1)} + \dfrac{11}{3(x - 2)}

Why It Matters

In calculus, integrals of rational functions are nearly impossible to evaluate directly but become straightforward after partial fraction decomposition. This technique also appears in solving differential equations and in Laplace transforms used in engineering.

Common Mistakes

Mistake: Applying the method when the fraction is improper (numerator degree ≥ denominator degree).

Correction: Always check degrees first. If the numerator's degree is not less than the denominator's, perform polynomial long division before decomposing the remainder.