Question

How do you use the rational root theorem to find the roots of `f(x) = x^3-4x^2-2x+8`?

Answer 1

Use the rational root theorem to identify the possible rational roots to try. Hence find `x = 4` and hence the other two irrational roots `x = +-sqrt(2)`

Explanation:

`f(x) = x^3-4x^2-2x+8`

By the rational root theorem, any rational roots of `f(x) = 0` must be expressible in lowest terms in the form `p/q` for some integers `p`, `q` where `p` is a divisor of the constant term `8` and `q` is a divisor of the coefficient `1` of the leading term.

So the only possible rational roots are:

`+-1`, `+-2`, `+-4`, `+-8`

`+-1` won't work since the leading term would be odd and the remaining terms are even.

`f(2) = 8-16-4+8 = -4`

`f(-2) = -8-16+4+8 = -12`

`f(4) = 64 - 64 - 8 + 8 = 0`

So `x = 4` is a root and `(x-4)` a factor.

`x^3-4x^2-2x+8 = (x-4)(x^2-2)`

The remaining roots are the irrational roots of `x^2-2 = 0`, that is:

`x = +-sqrt(2)`