# How do you factor 8x^3-12x^2+2x-3?

Sep 27, 2016

(8x^3-12x+2x-3=color(green)((2x-3)(4x^2+1))

#### Explanation:

Notice that the ratio of the coefficients of the terms $8 {x}^{3}$ and $2 x$: $8 : 2 = 4 : 1$
is the same as the ratio of the coefficients of the terms $- 12 {x}^{2}$ and $- 3$: $- 12 : - 3 = 4 : 1$

This hints hat we should group the original expression as
$\textcolor{w h i t e}{\text{XXX}} \left(\textcolor{red}{8 {x}^{3} + 2 x}\right) - \left(\textcolor{b l u e}{12 {x}^{2} + 3}\right)$

color(white)("XXX")=color(red)(2x(4x^2+1)))-color(blue)(3(4x^2+1))

then extracting the common factor $\left(4 {x}^{2} + 1\right)$
$\textcolor{w h i t e}{\text{XXX}} = \left(2 x - 3\right) \left(4 {x}^{2} + 1\right)$

(Note that since $4 {x}^{2} \ge 0$ for $\forall x \in \mathbb{R}$,
$\textcolor{w h i t e}{\text{XXX}} 4 {x}^{2} + 1$ can not be equal to $0$
$\textcolor{w h i t e}{\text{XXX}}$and $\left(4 {x}^{2} + 1\right)$ has no Real factors.