Answer
There are three positive roots and no negative roots of the polynomial among which two can be real in nature.
Work Step by Step
It can be observed from the given polynomial that it changes signs three times. Then, according to Descartes' Rule of Signs, the polynomial can have a maximum of 3 positive roots.
Now, substituting x with $-x$, we obtain:
$\begin{align}
& f\left( -x \right)=3{{\left( -x \right)}^{5}}-2{{\left( -x \right)}^{4}}-2{{\left( -x \right)}^{2}}+\left( -x \right)-1 \\
& f\left( -x \right)=-3{{\left( x \right)}^{5}}-2{{\left( x \right)}^{4}}-2{{\left( x \right)}^{2}}-\left( x \right)-1 \\
\end{align}$
Then, it can be seen that the polynomial remains negative throughout; hence, it has no negative roots.
Thus, the polynomial has three positive roots and no negative roots.
