# Enigmatic polynomial!

Algebra Level 5

$\large \left (\prod_{k=6}^{1729} P_k N_k + \prod_{k=6}^{1729} Z_k \right )\bmod {11}$

Let $$f_k(x) = 1729x^k + 8x^{k-1} - x^3 + kx - 1$$ for integer $$6\leq k\leq 1729$$.

We further define these
- $$P_k$$ as the number of maximum possible positive roots of $$f_k(x)$$.
- $$N_k$$ as the number of maximum possible negative roots of $$f_k(x)$$.
- $$Z_k$$ as the number of maximum possible non-real roots of $$f_k(x)$$.

Evaluate the modulo of the sum of the products given above.

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