Question

How do you factor `27x^3-8y^3`?

Answer 1

The factored polynomial is `(3x-2y)(9x^2+6xy+4y^2)`.

Explanation:

You can use the difference of cubes factoring pattern:

`color(red)a^3-color(blue)b^3quad=quad(color(red)a-color(blue)b)(color(red)a^2+color(red)acolor(blue)b+color(blue)b^2)`

Right now, our terms are `a^3=27x^3` and `b^3=8y^3`. Take the cube root of each to find that `a=3x` and `b=2y`. Now plug them into the difference of cubes:

`color(white)=27x^3-8y^3`

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

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

`=(color(red)(3x)-color(blue)(2y))(color(red)3^2color(red)x^2+color(purple)6color(red)xcolor(blue)y+color(blue)2^2color(blue)y^2)`

`=(color(red)(3x)-color(blue)(2y))(color(red)(9x)^2+color(purple)6color(red)xcolor(blue)y+color(blue)(4y)^2)`

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

This is factored all the way. Hope this helped!