Question

How do you factor the expression `x^9-27y^6`?

Answer 1

`x^9-27y^6 = (x^3-3y^2)(x^6+3x^3y^2+9y^4)`

Explanation:

Given:

`x^9-27y^6`

Note that both `x^9 = (x^3)^3` and `27y^6 = (3y^2)^3` are perfect cubes.

So we can factor the given expression as a difference of cubes:

`A^3-B^3 = (A-B)(A^2+AB+B^2)`

with `A=x^3` and `B=3y^2` as follows:

`x^9-27y^6 = (x^3)^3-(3y^2)^3`

`color(white)(x^9-27y^6) = (x^3-3y^2)((x^3)^2+(x^3)(3y^2)+(3y^2)^2)`

`color(white)(x^9-27y^6) = (x^3-3y^2)(x^6+3x^3y^2+9y^4)`