Polynomial always non-negative Is there an elegant way to demonstrate that (for example) $x^{2016}-1008x^2+1007\ge 0$ $\forall x\in \mathbb{R}$ ? I tried to write it as sum of squares, but I didn't succeed.

Move $-1008x^2$ to the right side and use $\mathrm{AM}\geq\mathrm{GM}$
$$\frac{x^{2016}+\underbrace{1+1+\ldots+1}_{1007}}{1008} \geq \sqrt[1008]{x^{2016} \underbrace{1 \cdot 1\cdots 1}_{1007}}  =x^2$$
