# How do you factor (x + y)^3 + (x - y)^3?

Jun 5, 2018

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

#### Explanation:

${\left(x + y\right)}^{3} + {\left(x - y\right)}^{3}$

Think about Pascal's Triangle to expand these terms:
$= \left({x}^{3} + 3 x {y}^{2} + 3 {x}^{2} y + {y}^{3}\right) + \left({x}^{3} + 3 x {y}^{2} - 3 {x}^{2} y - {y}^{3}\right)$

Combine like terms
$= 2 {x}^{3} + 6 x {y}^{2}$

Factor:
$= \left(2 x\right) \left({x}^{2} + 3 {y}^{2}\right)$

Jun 5, 2018

$2 x \left({x}^{2} + 3 {y}^{2}\right)$

#### Explanation:

$\text{this is a "color(blue)"sum of cubes}$

•color(white)(x)a^3+b^3=(a+b)(a^2-ab+b^2)

$\text{here "a=x+y" and } b = x - y$

$= \left(x + y + x - y\right) \left[{\left(x + y\right)}^{2} - \left(x + y\right) \left(x - y\right) + {\left(x - y\right)}^{2}\right]$

$= 2 x \left[{x}^{2} + 2 x y + {y}^{2} - {x}^{2} + {y}^{2} + {x}^{2} - 2 x y + {y}^{2}\right]$

$= 2 x \left({x}^{2} + 3 {y}^{2}\right)$