# How do you use the rational roots theorem to find all possible zeros of  P(x) = 2x^5 + 3x^4 + 2x^2 - 2?

Apr 25, 2016

We seek for rational roots within numbers $\frac{p}{q}$, where $p | {a}_{0}$ and $q | {a}_{n}$.
If $p | - 2$ then $p \in \left\{\pm 1 , \pm 2\right\}$.
If $q | 2$ then $q \in \left\{\pm 1 , \pm 2\right\}$.
Thus $\frac{p}{q} \in R = \left\{\pm 1 , \pm \frac{1}{2} , \pm 2\right\}$.

Now, if the polynomian $P \left(x\right)$ has any rational roots, they all are in $R$ so we proceed by trial and error:

$P \left(1\right) = 5 \ne 0$
$P \left(- 1\right) = 1 \ne 0$
$P \left(0.5\right) = - 1.25 \ne 0$
$P \left(- 0.5\right) = - 1.375 \ne 0$
$P \left(2\right) = 118 \ne 0$
$P \left(- 2\right) = - 10 \ne 0$

From this we can conclude that the polynomial $P \left(x\right)$ has no rational roots.