Not Just Vieta's Derivatives!

Algebra Level 5

P(x)=x6x5x3x2xQ(x)=x4x3x21\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 z1,z2,z3,z_{1},z_{2},z_{3},and z4z_{4} are the roots of Q(x)=0Q(x) = 0, find P(z1)+P(z2)+P(z3)+P(z4)P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4}).

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