# How do you express (3x) / (x-3)^2 in partial fractions?

Oct 24, 2016

$\frac{3 x}{x - 3} ^ 2 \Leftrightarrow \frac{3}{x - 3} + \frac{9}{x - 3} ^ 2$

#### Explanation:

Partial fractions of $\frac{3 x}{x - 3} ^ 2$ are in the form $\frac{A}{x - 3} + \frac{B}{x - 3} ^ 2$

i.e. $\frac{3 x}{x - 3} ^ 2 \Leftrightarrow \frac{A}{x - 3} + \frac{B}{x - 3} ^ 2$

or $\frac{3 x}{x - 3} ^ 2 \Leftrightarrow \frac{A \left(x - 3\right) + B}{x - 3} ^ 2 = \frac{A x - 3 A + B}{x - 3} ^ 2$

Comparing like terms $A = 3$ and $- 3 A + B = 0$ i.e. $B = 3 A = 9$

Hence $\frac{3 x}{x - 3} ^ 2 \Leftrightarrow \frac{3}{x - 3} + \frac{9}{x - 3} ^ 2$