Partial Fraction Decomposition

Types

When you have factors like $(x - a) (x - b) (x - c) ...$

Example: $\frac{3 x + 1}{( x - 1 ) ( x + 2 )} = \frac{A}{x - 1} + \frac{B}{x + 2}$

When you have factors like $(x - a)^{2} (x - b)^{3} ...$

Example: $\frac{x ^{2} + 1}{( x - 1 ) ^{2} ( x + 3 )} = \frac{A}{x - 1} + \frac{B}{( x - 1 ) ^{2}} + \frac{C}{x + 3}$

When you have quadratic factors that don’t factor over the reals, like $x^{2} + 4$ , $x^{2} + x + 1$ , etc.

Example: $\frac{7 x ^{2} - 18 x + 16}{( x ^{2} + 4 ) ( x - 6 )} = \frac{A x + B}{x ^{2} + 4} + \frac{C}{x - 6}$

When irreducible quadratics are raised to powers.

Example: $\frac{x ^{3} + 1}{( x ^{2} + 1 ) ^{2} ( x - 1 )} = \frac{A x + B}{x ^{2} + 1} + \frac{C x + D}{( x ^{2} + 1 ) ^{2}} + \frac{E}{x - 1}$

Primary Applications:

Integration - Many rational functions become much easier to integrate after partial fraction decomposition.
Inverse Laplace Transforms - Converting from frequency domain back to time domain in engineering/physics.
Solving Differential Equations - Particularly useful in finding particular solutions.
Signal Processing - Analyzing transfer functions in control systems.
Series Expansions - Breaking complex expressions into simpler components.

Irreducible quadratic: The discriminant $b^{2} - 4 a c < 0$ , so it has no real roots
“Sum of squares”: Forms like $x^{2} + k^{2}$ (where $k > 0$ ) are always irreducible
Repeated factors: Look for exponents $> 1$

Linear factor $(x - a)$ : contributes $\frac{A}{x - a}$
Repeated linear factor $(x - a)^{n}$ : contributes $\frac{A _{1}}{x - a} + \frac{A _{2}}{( x - a ) ^{2}} + ... + \frac{A _{n}}{( x - a ) ^{n}}$
Irreducible quadratic $(a x^{2} + b x + c)$ : contributes $\frac{A x + B}{a x ^{2} + b x + c}$
Repeated irreducible quadratic $(a x^{2} + b x + c)^{n}$ : contributes $\frac{A _{1} x + B _{1}}{a x ^{2} + b x + c} + \frac{A _{2} x + B _{2}}{( a x ^{2} + b x + c ) ^{2}} + ... + \frac{A _{n} x + B _{n}}{( a x ^{2} + b x + c ) ^{n}}$