# How do you write the partial fraction decomposition of the rational expression  (x^2 - 3x) / ((x-1)(x+2))?

Jan 15, 2016

$\frac{- \frac{2}{3}}{x - 1} - \frac{\frac{10}{3}}{x + 2}$

#### Explanation:

let ( x^2 - 3x) /((x - 1 )(x + 2 )) ≣ A/(x - 1) + B/(x + 2 )

Since the factors on the denominator are of degree 1 (linear) then the numerators will be constants (degree 0 ) denoted by A and B .

Multiply both sides of the equation by (x - 1 )(x + 2 ) :

$\Rightarrow {x}^{2} - 3 x = A \left(x + 2\right) + B \left(x - 1\right) \ldots \ldots \ldots \ldots \ldots \ldots \textcolor{red}{\left(\cdot\right)}$

(Note that if x = 1 then the term in B will be 0. Similarly if x = - 2 then the term in A will also be 0 . )

let x = 1 and substitute in equation $\textcolor{red}{\left(\cdot\right)}$

$\Rightarrow - 2 = 3 A \Rightarrow A = - \frac{2}{3}$

let x = -2 and substitute in equation $\textcolor{red}{\left(\cdot\right)}$

$\Rightarrow 10 = - 3 B \Rightarrow B = - \frac{10}{3}$

Finally :

$\frac{{x}^{2} - 3 x}{\left(x - 1\right) \left(x + 2\right)} = \frac{- \frac{2}{3}}{x - 1} - \frac{\frac{10}{3}}{x - 2}$