AMC 12 2004a Problem 23

A polynomial

$$P(x) = c_{2004}x^{2004} + c_{2003}x^{2003} + ... + c_1x + c_0$$

has real coefficients with $c_{2004}\not = 0$ and $2004$ distinct complex zeroes $z_k = a_k + b_ki$, $1\leq k\leq 2004$ with $a_k$ and $b_k$ real, $a_1 = b_1 = 0$, and

$$\sum_{k = 1}^{2004}{a_k} = \sum_{k = 1}^{2004}{b_k}.$$

Which of the following quantities can be a nonzero number?

$\text {(A) } c_0 \text {(B) } c_{2003} \text {(C) } b_2b_3...b_{2004} \text {(D) } \sum_{k = 1}^{2004}{a_k} \text {(E) }\sum_{k = 1}^{2004}{c_k}$