<s>[INST] Questions involving two polynomials Let $$P(x)=x^{6}-x^{5}-x^{3}-x^{2}-x$$
$$Q(x)=x^{4}-x^{3}-x^{2}-1$$
Let $z_{1}, z_{2}, z_{3}, z_{4}$ be the roots of Q(x)
Prove that $$P\left (z_{1}  \right )+ P\left (z_{2}  \right )+ P\left (z_{3}  \right )+ P\left (z_{4}  \right )=6$$
 [/INST] HINT:
So, $$z_1^4-z_1^3-z_1^2-1=0\ \ \ \ \ (1)$$
$$P(z_1)=z_1^6-z_1^5-z_1^3-z_1^2-z_1$$
$$=z_1^2(z_1^4-z_1^3-z_1^2-1)+z_1^4-z_1^3-z_1=z_1^2-z_1+1$$
Let $z_1^2-z_1+1=y$
$$z_1^4-z_1^3-z_1^2-1=z_1^2(z_1^2-z_1+1)-2z_1^2-1=z_1^2y-2(y-1+z_1)-1$$
$$=(y+z_1-1)y-2(y-1+z_1)-1$$
$$\implies (y+z_1-1)y-2(y-1+z_1)-1=0$$
Express $z_1$ in terms of $y$ and put the value in $(1)$
Then use Vieta's formulas 1,2,3
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