# Question #1d5ec

Dec 18, 2017

There are no rational zeroes.

#### Explanation:

The rational root theorem says that the possible rational zeroes can be found by dividing plus or minus the factors of the last coefficient with plus or minus the factors of the first coefficient. The prime factors of $10$ is $5 \cdot 2$ and the prime factors of $1$ are just $1$.

So, our possible solutions are:
$\pm 5 , \pm 2 , \pm 1$

If we plug these values into our function, we see that none of them make the function equal to $0$, so we have no rational roots.

If you're curious, there is an irrational solution, but it is rather complicated to obtain...
$x = \frac{1}{3} \left(4 + \sqrt[3]{181 - 6 \sqrt{849}} + \sqrt[3]{181 + 6 \sqrt{849}}\right)$