# How do you factor  125x^3+64?

Dec 14, 2015

Use the sum of cubes identity to find:

$125 {x}^{3} + 64 = \left(5 x + 4\right) \left(25 {x}^{2} - 20 x + 16\right)$

#### Explanation:

Both $125 {x}^{3} = {\left(5 x\right)}^{3}$ and $64 = {4}^{3}$ are perfect cubes, so this is a natural case for the sum of cubes identity:

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

With $a = 5 x$ and $b = 4$ we find:

$125 {x}^{3} + 64$

$= {\left(5 x\right)}^{3} + {4}^{3}$

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

$= \left(5 x + 4\right) \left(25 {x}^{2} - 20 x + 16\right)$