Let $f=X^{10}-5X^{8}+a_{7}X^{7}+ a_6X^6+...+a_{1}X+a_{0}$ be a polynomial with real coefficients. We are asked to determine its roots $x_{1},x_{2},...,x_{10}$, where $|x_{1}|\geq 3$ and $x_{1},x_{2},...,x_{10}\in \mathbb{R}$.
I know that $x_{1}+...+x_{10}=0$, $x_{1}x_{2}+...+x_{9}x_{10}=-5$ and $x_{2}^{2}+...+x_{10}^{2}\leq 1$, but I haven't found anything meaningful besides this yet.
Thank you!