How do you factor 8x^3y^6-125?

Jan 3, 2016

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

Explanation:

This is a difference of cubes, which takes the general form:

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

Recognize that $8 {x}^{3} {y}^{6} = {\left(2 x {y}^{2}\right)}^{3}$ and that $125 = {5}^{3}$.

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

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