Question

How do you factor `w^3-3w^2-9w+27`?

Answer 1

`(w^3-3w^2-9w+27)=color(blue)((x-3)(x-3)(x+3))`

Explanation:

Writing `w^3-3w^2-9w+27`
as
`color(white)("XXX")w^3-3w^2-3^3w+3^3`
makes it fairly clear that
`color(white)("XXX")w=3` is a zero of this expression.

That is `(w-3)` is a factor of `(w^3-3w^2-9w+27)`

If we divided `(w^3-3w^2-9w+27)` by `(w-3)`
(either by polynomial long division or synthetic division)
we find
`color(white)("XXX")(x^2-9)` is also a factor

...but `(x^2-9)=(x^2-3^2)=(x-3)(x+3)`

So a complete factoring is
`color(white)("XXX")(w^3-3w^2-9w+27)=(x-3)(x-3)(x+3)`