Question 2
For every positive integer $n$, prove that the polynomial
$$P_n(x)=\sum_{k=0}^n2^k\binom{2n}{2k}x^k(x-1)^{n-k}$$
has exactly $n$ distinct real roots.
$$P_n(x)=\sum_{k=0}^n2^k\binom{2n}{2k}x^k(x-1)^{n-k}$$
has exactly $n$ distinct real roots.