Partial Fraction Decomposition

When to Use Partial Fractions

P(x)/Q(x) → A/(x−a) + B/(x−b) + (Cx+D)/(x²+bx+c) + …

Partial fraction decomposition applies when P(x)/Q(x) is a proper rational function — meaning deg P < deg Q. If the fraction is improper, use polynomial long division first, then decompose the remainder.

Main uses:

Preparing for integration in Calculus (∫ P/Q dx)
Simplifying complex rational expressions
Inverse Laplace transforms in Differential Equations

The denominator Q(x) must be fully factored into linear and irreducible quadratic factors over the reals before you can set up the decomposition.

Factor Types & Their Forms

Distinct linear factor (x − a):
Contributes A / (x − a). Find A using the cover-up (Heaviside) method: multiply both sides by (x − a) and set x = a.

Repeated linear factor (x − a)²:
Contributes A / (x − a) + B / (x − a)². Find B by cover-up at x = a; find A by substituting a second value or matching coefficients.

Irreducible quadratic factor (x² + bx + c), discriminant < 0:
Contributes (Ax + B) / (x² + bx + c). Solve a linear system by expanding and matching polynomial coefficients.

A quadratic x²+bx+c is irreducible over the reals when its discriminant b²−4c < 0 (no real roots).

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