Question

How do you factor `x^4-2x^3-12x^2+18x+27`?

Answer 1

Use the rational roots theorem to find:

`x^4-2x^3-12x^2+18x+27 = (x+1)(x+3)(x-3)(x-3)`

Explanation:

Let `f(x) = x^4-2x^3-12x^2+18x+27`

By the rational roots theorem, any rational roots of `f(x) = 0` must be of the form `p/q` where `p, q` are integers, `q != 0`, `p` a divisor of `27` and `q` a divisor of `1`.

So the only possible rational roots are:

`+-1`, `+-3`, `+-9`, `+-27`

`f(1) = 1-2-12+18+27 = 32`
`f(-1) = 1+2-12-18+27 = 0`
`f(3) = 81-54-108+54+27 = 0`
`f(-3) = 81+54-108-54+27 = 0`

So `x=-1`, `x=3` and `x=-3` are roots of `f(x) = 0` and `(x+1)`, `(x-3)` and `(x+3)` are factors of `f(x)`.

The remaining factor must be `(x-3)` in order that when multiplied by the other factors the coefficient of the `x^4` term is `1` and the constant term `27`.

graph{x^4-2x^3-12x^2+18x+27 [-10, 10, -5, 5]}