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.

This is a quartic, its discriminant tells all you need.
