# Not Just Vieta's Derivatives!

Algebra Level 5

\begin{aligned} P(x) &=& x^{6} - x^{5} - x^{3} - x^{2} - x \\ Q(x) &=& x^{4} - x^{3} - x^{2} - 1 \\ \end{aligned}

Consider the polynomials above.

Given that $z_{1},z_{2},z_{3},$and $z_{4}$ are the roots of $Q(x) = 0$, find $P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4})$.

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