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

##### 1 Answer
Sep 22, 2015

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 ${x}^{3} + 2 x - 9 = 0$ will be expressible in the form $\frac{p}{q}$ in lowest terms, where $p , q \in \mathbb{Z}$, $q \ne 0$, $p$ 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:

$\pm 1$, $\pm 3$, $\pm 9$

None of these work.

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

$x = \sqrt[3]{\frac{81 + \sqrt{6657}}{18}} + \sqrt[3]{\frac{81 - \sqrt{6657}}{18}} \approx 1.762496$