proving zeros of a polynomial are not real I'm working on a optimization problem and need to show that 
\begin{equation}
\frac{1}{2}x^4 - x^3 -x + 100 = 0
\end{equation} has no real solution in order to prove certain properties about the function I am working on.  I don't need to know the zeros just show that they don't exist.  I'm doing this because I would like to show that 
\begin{equation}
\frac{1}{2}x^4 - x^3 > x - 100
\end{equation}
 for all $x$.  What would be the best way to go about this?
Thanks in advance.

We can do it by looking at the expression for different ranges of $x$. It is clearly positive if $x\le 0$. 
If $0 < x \le 4$, then $x^3+x\lt 100$, so our expression is positive even without the aid of $\frac{1}{2}x^4$. If $x\gt 4$, then $\frac{1}{2}x^4\gt 2x^3$, so $\frac{1}{2}x^4-x^3-x\gt 0$.
