Partial fraction decomposition of $\frac{x-1}{x^3+x^2}$

Question

Why the partial fraction expansion of $\frac{x-1}{x^3+x^2}$ is

$$\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1}$$

if I only have that $x^3 + x^2$ is the same as $x^2(x + 1)$?

Answer 1

So you have $$\frac{x-1}{x^2(1+x)}=\frac {Ax+B}{x^2}+\frac C{x+1}=\frac Ax+\frac B{x^2}+\frac C{x+1}$$

The second form is often preferred for practical reasons - e.g. the first term integrates to a logarithm and the second doesn't.

Answer 2

Since you are looking for an summation of simple fractions it must be something like $$\frac{ax+b}{x^2}+\frac{c}{x+1}$$ It follows $\frac{ax+b}{x^2}$ also can be written as $$\frac{ax}{x^2}+\frac{b}{x^2}\quad\text{ or }\quad \frac{a}{x}+\frac{b}{x^2}$$