# How do you factor 2x^3-3x^2-10x+15?

Jul 16, 2017

$2 {x}^{3} - 3 {x}^{2} - 10 x + 15 = \left({x}^{2} - 5\right) \left(2 x - 3\right)$

$\textcolor{w h i t e}{2 {x}^{3} - 3 {x}^{2} - 10 x + 15} = \left(x - \sqrt{5}\right) \left(x + \sqrt{5}\right) \left(2 x - 3\right)$

#### Explanation:

Given:

$2 {x}^{3} - 3 {x}^{2} - 10 x + 15$

Note that the ratio of the first and second terms is the same as that of the third and fourth terms. So this cubic will factor by grouping:

$2 {x}^{3} - 3 {x}^{2} - 10 x + 15 = \left(2 {x}^{3} - 3 {x}^{2}\right) - \left(10 x - 15\right)$

$\textcolor{w h i t e}{2 {x}^{3} - 3 {x}^{2} - 10 x + 15} = {x}^{2} \left(2 x - 3\right) - 5 \left(2 x - 3\right)$

$\textcolor{w h i t e}{2 {x}^{3} - 3 {x}^{2} - 10 x + 15} = \left({x}^{2} - 5\right) \left(2 x - 3\right)$

$\textcolor{w h i t e}{2 {x}^{3} - 3 {x}^{2} - 10 x + 15} = \left({x}^{2} - {\left(\sqrt{5}\right)}^{2}\right) \left(2 x - 3\right)$

$\textcolor{w h i t e}{2 {x}^{3} - 3 {x}^{2} - 10 x + 15} = \left(x - \sqrt{5}\right) \left(x + \sqrt{5}\right) \left(2 x - 3\right)$