Question

How do you factor `x^3+8y^3`?

Answer 1

`x^3+8y^3 = (x+2y)(x^2-2xy+4y^2)`

Explanation:

In general:
`color(white)("XXX")(color(red)(a)^3+color(blue)(b)^3) = (color(red)(a)+color(blue)(b))(color(red)(a)^2-color(red)(a)color(blue)(b)+color(blue)(b)^2)`

for various proofs of this see:
http://www.qc.edu.hk/math/Junior%20Secondary/sum%20of%20two%20cubes.htm

`(x^3+8y^3) = (color(red)(x)^3+color(blue)((2y))^3)`

Substituting
`color(white)("XXX")color(red)(x)` for `color(red)(a)`
and
`color(white)("XXX")color(blue)(2y)` for `color(blue)(b)`
in the general form
`color(white)("XXX")`gives the answer above.