Partial Fraction Decomposition: Complete Guide
Partial fraction decomposition is the algebraic operation of reversing common-denominator fraction addition. It expresses a complex rational function as a sum of simpler elementary fractions.
The Mathematical Setup
For a proper rational function with distinct linear denominator factors:
\frac{px + q}{(x - r_1)(x - r_2)} = \frac{A}{x - r_1} + \frac{B}{x - r_2}
The Heaviside Cover-Up Method
Oliver Heaviside developed an elegant shortcut for finding coefficients of distinct linear factors without solving full linear systems:
- 1To solve for $A$: Cover up the $(x - r_1)$ factor in the denominator and substitute $x = r_1$:
A = \left. \frac{px + q}{x - r_2} \right|_{x = r_1} = \frac{p r_1 + q}{r_1 - r_2}
- 1To solve for $B$: Cover up the $(x - r_2)$ factor and evaluate at $x = r_2$:
B = \left. \frac{px + q}{x - r_1} \right|_{x = r_2} = \frac{p r_2 + q}{r_2 - r_1}
Essential Rules by Factor Type
| Factor Type in Denominator | Assumed Partial Fraction Form |
|---|---|
| Distinct Linear: $(x - r)$ | $\frac{A}{x - r}$ |
| Repeated Linear: $(x - r)^2$ | $\frac{A}{x - r} + \frac{B}{(x - r)^2}$ |
| Irreducible Quadratic: $(x^2 + ax + b)$ | $\frac{Ax + B}{x^2 + ax + b}$ |