There exists a polynomial $P$ of degree 5 with the following property: If $z$ is a complex number such that $z^5 + 2004z = 1,$ then $P(z^2) = 0.$  Calculate
\[\frac{P(1)}{P(-1)}.\]
Question:
There exists a polynomial $P$ of degree 5 with the following property: If $z$ is a complex number such that $z^5 + 2004z = 1,$ then $P(z^2) = 0.$  Calculate
\[\frac{P(1)}{P(-1)}.\]
Answer:
Let's think step by step.

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-\frac{2010012}{2010013}
There exists a polynomial $P$ of degree 5 with the following property: If $z$ is a complex number such that $z^5 + 2004z = 1,$ then $P(z^2) = 0.$  Calculate
\[\frac{P(1)}{P(-1)}.\]
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