# What are the possible rational zeros of f(x) = 2x^3+5x^2-8x-10 ?

Jul 15, 2017

The possible rational zeros are:

$\pm \frac{1}{2} , \pm 1 , \pm 2 , \pm \frac{5}{2} , \pm 5 , \pm 10$

#### Explanation:

Given:

$f \left(x\right) = 2 {x}^{3} + 5 {x}^{2} - 8 x - 10$

By the rational roots theorem, any rational zeros of $f \left(x\right)$ are expressible in the form $\frac{p}{q}$ for integers $p , q$ with $p$ a divisor of the constant term $- 10$ and $q$ a divisor of the coefficient $2$ of the leading term.

That means that the only possible rational zeros are:

$\pm \frac{1}{2} , \pm 1 , \pm 2 , \pm \frac{5}{2} , \pm 5 , \pm 10$

In practice, none of these are zeros of $f \left(x\right)$, so it has no rational zeros.