Answer
$$ - \frac{9}{x} + \frac{9}{{x - 1}}$$
Work Step by Step
$$\eqalign{
& \frac{9}{{x\left( {x - 3} \right)}} \cr
& {\text{The partial fraction decomposition is }} \cr
& \frac{9}{{x\left( {x - 3} \right)}} = \frac{A}{x} + \frac{B}{{x - 3}} \cr
& {\text{Multiply each side by }}\left( {x - 3} \right)\left( {x + 1} \right) \cr
& 9 = A\left( {x - 3} \right) + Bx\,\,\,\,\,\,\,\,\,\,\left( 1 \right) \cr
& \cr
& {\text{Let }}x = 0{\text{ into the equation }}\left( 1 \right) \cr
& 9 = A\left( {0 - 3} \right) + B\left( 0 \right)\, \cr
& 9 = - 3A \cr
& A = - 3 \cr
& {\text{Let }}x = 3{\text{ into the equation }}\left( 1 \right) \cr
& 9 = A\left( {3 - 1} \right) + B\left( 3 \right) \cr
& B = 3 \cr
& \cr
& {\text{Use }}A{\text{ and }}B{\text{ to find the partial fraction decomposition}} \cr
& \frac{9}{{x\left( {x - 3} \right)}} = - \frac{9}{x} + \frac{9}{{x - 1}} \cr} $$
