Polynomials going crazy!

Algebra Level 5

The polynomial \(f(x) = x^{2007} + 17x^{2006} + 1 \) has distinct roots \(r_1, r_2,\ldots, r_{2007} \). A polynomial \(P\) of degree 2007 has the property that \(P \left( r_j + \dfrac1{r_j} \right) = 0 \) for \(j = 1,2,\ldots,2007 \).

Compute \( \dfrac{259}{17} \times \dfrac{P(1)}{P(-1)} \).

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