# How do you simplify \frac { x ^ { 3} - 4x } { x ^ { 2} + x - 2}?

Mar 14, 2018

$\frac{x \left(x - 2\right)}{x - 1}$

#### Explanation:

$\text{Factorise numerator/denominator}$

• " numerator "x^3-4x

$\text{take out a "color(blue)"common factor of x}$

$= x \left({x}^{2} - 4\right)$

${x}^{2} - 4 \text{ is a "color(blue)"difference of squares}$

•color(white)(x)a^2-b^2=(a-b)(a+b)

$\Rightarrow {x}^{3} - 4 x = x \left(x - 2\right) \left(x + 2\right)$

• " denominator "x^2+x-2

$\text{the factors of - 2 which sum to + 1 are + 2 and - 1}$

$\Rightarrow {x}^{2} + x - 2 = \left(x + 2\right) \left(x - 1\right)$

$\Rightarrow \frac{{x}^{3} - 4 x}{{x}^{2} + x - 2}$

$= \frac{x \left(x - 2\right) \cancel{\left(x + 2\right)}}{\cancel{\left(x + 2\right)} \left(x - 1\right)}$

$= \frac{x \left(x - 2\right)}{x - 1}$

$\text{with restriction } x \ne 1$