Question

How do you use the rational root theorem to find the roots of `x^3+2x-9=0`?

Answer 1

The rational root theorem will only tell you what the possible rational roots are. This cubic has no rational roots.

Explanation:

By the rational root theorem, any rational root of `x^3+2x-9=0` will be expressible in the form `p/q` in lowest terms, where `p, q in ZZ`, `q != 0`, `p` a divisor of the constant term `9` and `q` a divisor of the coefficient `1` of the leading term.

So the possible rational roots are:

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

None of these work.

Using Cardano's method, I found one Real root:

`x = root(3)((81+sqrt(6657))/18) + root(3)((81-sqrt(6657))/18) ~~ 1.762496`