Question

How do you factor `27x ^3 + 216`?

Answer 1

`27x^3+216 = 27(x+2)(x^2-2x+4)`

Explanation:

Both terms in the original expression have a factor of `27` which can be extracted as a factor of the expression.

The remaining factor `(x^3+8)=(x^3+2^3)` is of the form
`color(white)("XXX")(x^3+a^3)`
which we know has a factors `(x+a)` and `(x^2-ax+a^2)`
or in this case
`color(white)("XXX")(x^3+8)=(x+2)(x^2-2x+4)`

By checking the discriminant of `(x^2-2x+4)`
`color(white)("XXX")(-2)^2+4(1)(4) < 0 rArr` no further Real roots