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.
 A: This is a quartic, its discriminant tells all you need.
A: 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 $\frac{1}{2}x^4-x^3-x\gt 0$.
A: For negative $x$, all summands are $\ge 0$ so $f(x)\ge 100$.
For $0\le x \le 4$, $f(x)\ge 100-x^3-x\ge 100-64-4>0$. 
Note that 
$$ f(x+1)=\frac12x^4+x^3-2x+\frac{197}2\ge (x^2-2)x+\frac{197}2>0$$
for $x>\sqrt 2$, so $f(x)>0$ for $x>\sqrt 2-1$. This covers all of $\mathbb R$.
A: Find the derivative ov the left side of your first equation and use it's zeros to find the minima of that function. They will, hopefully, all be positive :)
A: Sketch $\frac{1}{2}x^4 - x^3 -x + 100$. There are two changes in concavity between $x=-2$ and $x=3$, so the graph can't "change direction" outside those values, and the minimum must be between them. Direct calculation shows that within that interval, the function exceeds 50 at each integer-valued $x$.
For there to be a zero within $0.5$ units of any of these integer points, the function would have to have an average rate of change of more than $50$ on an interval of length less than $0.5$, and by the mean value theorem, the absolute value of the derivative would have to exceed $\frac{50}{0.5}=100$ somewhere. It doesn't. The absolute value of the derivative in that interval is easily bounded above by $2|3|^3+3|3|^2+1=82$ (the sum of the absolute values of the terms of the derivative at the largest $|x|$-value.
