# How do you factor x^3+6x^2-5x-30?

Apr 15, 2018

$x = - 6 , x =$$\pm$sqrt5

#### Explanation:

rearrange the equation
${x}^{3} - 5 x + 6 {x}^{2} - 30$

take out the common factors
$x \left({x}^{2} - 5\right) + 6 \left({x}^{2} - 5\right)$

compress the terms
$\left(x + 6\right) \left({x}^{2} - 5\right)$

solve for $x$
$x = - 6 , x =$$\pm$sqrt5

Apr 15, 2018

$\left(x + 6\right) \left(x - \sqrt{5}\right) \left(x + \sqrt{5}\right)$

#### Explanation:

$\textcolor{b l u e}{\text{factor by grouping the terms}}$

$= \textcolor{red}{{x}^{2}} \left(x + 6\right) \textcolor{red}{- 5} \left(x + 6\right)$

$\text{take out the "color(blue)"common factor } \left(x + 6\right)$

$= \left(x + 6\right) \left(\textcolor{red}{{x}^{2} - 5}\right)$

${x}^{2} - 5 \text{ can be factored using "color(blue)"difference of squares}$

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

${x}^{2} - 5 = {x}^{2} - {\left(\sqrt{5}\right)}^{2}$

$\text{with "a=x" and } b = \sqrt{5}$

$\Rightarrow {x}^{2} - 5 = \left(x - \sqrt{5}\right) \left(x + \sqrt{5}\right)$

$\Rightarrow {x}^{3} + 6 {x}^{2} - 5 x - 30 = \left(x + 6\right) \left(x - \sqrt{5}\right) \left(x + \sqrt{5}\right)$