# How do you factor 16x^3 - 250?

Sep 27, 2015

Separate out the common scalar factor $2$, then use the difference of cubes identity to find:

$16 {x}^{3} - 250 = 2 \left(2 x - 5\right) \left(4 {x}^{2} + 10 x + 25\right)$

#### Explanation:

Both $16$ and $250$ are divisible by $2$ so separate that out first.

$16 {x}^{3} - 250 = 2 \left(8 {x}^{3} - 125\right)$

Now $8 = {2}^{3}$ and $125 = {5}^{3}$

So we find:

$2 \left(8 {x}^{3} - 125\right)$

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

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

$= 2 \left(2 x - 5\right) \left(4 {x}^{2} + 10 x + 25\right)$

...using the difference of cubes identity:

${a}^{3} - {b}^{3} = \left(a - b\right) \left({a}^{2} + a b + {b}^{2}\right)$