Question 17
Prove that for all positive integers $n$, all complex roots $r$ of the polynomial
$$
P(x) = (2 n) x^{2 n} + (2 n-1) x^{2 n-1} + \cdots + (n+1) x^{n+1} + n x^{n} + (n+1) x^{n-1} + \cdots + (2 n-1) x + 2 n
$$
lie on the unit circle (i.e. $|r|=1$).
$$
P(x) = (2 n) x^{2 n} + (2 n-1) x^{2 n-1} + \cdots + (n+1) x^{n+1} + n x^{n} + (n+1) x^{n-1} + \cdots + (2 n-1) x + 2 n
$$
lie on the unit circle (i.e. $|r|=1$).