Polynomial Zero Sum Property
A polynomial \[ P(x)=c_{2004}x^{2004}+ c_{2003}x^{2003}+ \cdots+ c_{1}x+ c_{0} \]has real coefficients with $ c_{2004} \neq 0 $ and 2004 distinct complex zeros $ z_{k}=a_{k}+ b_{k}i $, $ 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? A) $ c_0 $. B) $ c_{2003} $. C) $ b_{2}b_{3} \dotsm b_{2004} $. D) $ \sum_{k=1}^{2004}a_{k} $. E) $ \sum_{k=1}^{2004}c_{k} $.
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