Enigmatic polynomial!

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

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.

This problem is original and is created by me.