# How do you factor x^3 - x^2 - 2x + 2=0?

Mar 11, 2018

$x = \sqrt{2} , x = - \sqrt{2} , x = 1$

#### Explanation:

Rewrite the equation
${x}^{3} - {x}^{2} - 2 x + 2 = 0$
${x}^{2} \left(x - 1\right) - 2 \left(x - 1\right) = 0$
$\left({x}^{2} - 2\right) \left(x - 1\right) = 0$
${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$
${x}^{2} - 2 = {x}^{2} - {\left(\sqrt{2}\right)}^{2} = \left(x - \sqrt{2}\right) \left(x + \sqrt{2}\right)$
$\left(x - \sqrt{2}\right) \left(x + \sqrt{2}\right) \left(x - 1\right) = 0$
$\left(x - \sqrt{2}\right) = 0 , \left(x + \sqrt{2}\right) = 0 , \left(x - 1\right) = 0$
$x = \sqrt{2} , x = - \sqrt{2} , x = 1$

Mar 11, 2018

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

#### Explanation:

$\text{note that the coefficients } 1 - 1 - 2 + 2 = 0$

$\Rightarrow \left(x - 1\right) \text{ is a factor}$

$\text{divide the polynomial by } \left(x - 1\right)$

$\textcolor{red}{{x}^{2}} \left(x - 1\right) \cancel{\textcolor{m a \ge n t a}{+ {x}^{2}}} \cancel{- {x}^{2}} - 2 x + 2$

$= \textcolor{red}{{x}^{2}} \left(x - 1\right) \textcolor{red}{- 2} \left(x - 1\right) \cancel{\textcolor{m a \ge n t a}{- 2}} \cancel{+ 2}$

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

${x}^{2} - 2 \leftarrow \textcolor{b l u e}{\text{is a difference of squares}}$

â€¢color(white)(x)a^2-b^2=(a-b)(a+b)

$\text{with "a=x" and } b = \sqrt{2}$

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

$\Rightarrow {x}^{3} - {x}^{2} - 2 x + 2 = 0$

$\Rightarrow \left(x - 1\right) \left(x - \sqrt{2}\right) \left(x + \sqrt{2}\right) = 0$

$\Rightarrow x = 1 \text{ or } x = \pm \sqrt{2}$