Consider the polynomials P(x)=x^{6}-x^{5}-x^{3}-x^{2}-x and Q(x)=x^{4}-x^{3}-x^{2}-1. Given that z_{1}, z_{2}, z_{3}, and z_{4} are the roots of Q(x)=0, find P\left(z_{1}\right)+P\left(z_{2}\right)+ P\left(z_{3}\right)+P\left(z_{4}\right).
Consider the polynomials P(x)=x^{6}-x^{5}-x^{3}-x^{2}-x and Q(x)=x^{4}-x^{3}-x^{2}-1. Given that z_{1}, z_{2}, z_{3}, and z_{4} are the roots of Q(x)=0, find P\left(z_{1}\right)+P\left(z_{2}\right)+ P\left(z_{3}\right)+P\left(z_{4}\right).