Not Just Vieta's Derivatives!

Algebra Level 5

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

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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