Question 85
Let $z = \cos \frac{2\pi}{2011} + i \sin \frac{2\pi}{2011}$, and let
$$
P(x) = x^{2008} + 3 x^{2007} + 6 x^{2006} + \ldots + \frac{2008 \cdot 2009}{2} x + \frac{2009 \cdot 2010}{2}
$$
for all complex numbers $x$. Evaluate $P(z) P\left(z^{2}\right) P\left(z^{3}\right) \ldots P\left(z^{2010}\right)$.
$$
P(x) = x^{2008} + 3 x^{2007} + 6 x^{2006} + \ldots + \frac{2008 \cdot 2009}{2} x + \frac{2009 \cdot 2010}{2}
$$
for all complex numbers $x$. Evaluate $P(z) P\left(z^{2}\right) P\left(z^{3}\right) \ldots P\left(z^{2010}\right)$.