For $7$, Let $x=a_{2016}+a_{2014} +...+a_0$ and $b=a_{2015}+a_{2013}+...+a_1$, suppose that $P(1)P(-1)\geq 0$
Give us $(x+y)(x-y)\geq 0$, so $|x|\geq |y|$ contradiction, so $P(1)P(-1)<0$
Suppose $-1<r_1<r_2<...<r_{2t}<1$ are root of $P(x)$, then IMV give us $P(1)$ and $P(-1)$ have same sign, contradiction, this finish the prove
|