Example Problems -Partial Fraction Decomposition

Introduction

Rewrite the following rational expressions as a sum of multiple rational expressions with lower-degree polynomials in the deonomintor.

1. $$\frac{x+6}{x^2 - 7x + 12}$$

Answer:

$$ \displaylines{ \frac{x+6}{x^2 - 7x + 12} = \frac{x+6}{(x-3)(x-4)} \\ \\ \frac{x+6}{(x-3)(x-4)} = \boxed{\frac{A}{x-3} + \frac{B}{x-4}} } $$

2. $$\frac{x^2 + 4x - 2}{(x^2 + 4)(x-2)}$$

Answer:

$$ \frac{x^2 + 4x - 2}{(x^2 + 4)(x+1)} = \boxed{\frac{Ax + B}{x^2 + 4} + \frac{C}{x+1}} $$

3. $$\frac{x-\pi}{x(x^2+1)(x+5)^2} $$

   Answer:

   $$\frac{x-\pi}{x(x^2+1)(x+5)^2} = \boxed{\frac{A}{x} + \frac{Bx+C}{x^2+1} + \frac{D}{(x+5)} + \frac{E}{(x+5)^2}}$$