Question 118
For every positive integer $n$, prove that the polynomial
$$
P_n(x) = \sum_{k=0}^{n} 2^k \binom{2n}{2k} x^k (x-1)^{n-k}
$$
has $n$ real roots.
(Note: $\binom{2n}{2k}$ is a binomial coefficient.)
$$
P_n(x) = \sum_{k=0}^{n} 2^k \binom{2n}{2k} x^k (x-1)^{n-k}
$$
has $n$ real roots.
(Note: $\binom{2n}{2k}$ is a binomial coefficient.)