Afternonz wrote:
Let $P(x)$ be a polynomial with real coefficients. Prove that not all roots of $x^3P(x)+1$ are real.
The four lowest degree summands of $x^3P(x)+1$ are $P(0)x^3+1$ and so $ 0$ is not a root and $\sum\frac1{r_i}=0$ and $\sum\frac1{r_i^2}=0$, impossible if all $r_i\in\mathbb R$
Q.E.D.