# How do you factor 9(x + 2y + z)^2 - 16(x - 2y + z)^2?

Jul 4, 2015

This is a difference of squares.

#### Explanation:

Difference of squares: ${u}^{2} - {v}^{2} = \left(u + v\right) \left(u - v\right)$

$9 {\left(x + 2 y + z\right)}^{2} = {\left(3 \left(x + 2 y + z\right)\right)}^{2}$ That will be our ${u}^{2}$

$16 {\left(x - 2 y + z\right)}^{2} = {\left(4 \left(x - 2 y + z\right)\right)}^{2}$. That will be our ${v}^{2}$

$9 {\left(x + 2 y + z\right)}^{2} - 16 {\left(x - 2 y + z\right)}^{2} = {\left(3 \left(x + 2 y + z\right)\right)}^{2} - {\left(4 \left(x - 2 y + z\right)\right)}^{2}$

$= \left[3 \left(x + 2 y + z\right) + 4 \left(x - 2 y + z\right)\right] \left[3 \left(x + 2 y + z\right) - 4 \left(x - 2 y + z\right)\right]$

Each of these factors can be simplified, to get:

$\left(7 x - 2 y + 7 z\right) \left(- x + 14 y - z\right)$