Find $z_1^5 + z_2^5 + z_3^5$ given that $z_1$, $z_2$, and $z_3$ are the roots of $z^3 -z -1 = 0$ Hello everyone I have the polynomial $z^3 -z -1 = 0$ and $z_1 , z_2 , z_3$ are the roots of this polynomial.

How can I find $z_1^5 + z_2^5 + z_3^5 ?$

I know that $z_1 + z_2 + z_3 = 0 , z_1 \cdot z_2 + z_1 \cdot z_3 + z_2 \cdot z_3 = -1 , z_1z_2z_3 = 1$
Here is a simple slightly tedious method:

Let $A=\begin{bmatrix} 0 &amp; 0 &amp; 1 \\ 1 &amp; 0 &amp; 1 \\ 0 &amp; 1 &amp; 0 \end{bmatrix}$ and note 
that $\chi_A(z) = z^3-z-1$. Hence the sum of the $5$th powers of the roots is given by
$\operatorname{tr} A^5 = 5$.
Your equation is of the form:
$$y^3+Ay=B$$
where:
$$y=z,A=-1,B=1$$
Now substitute:
$$3st=A$$
$$s^3-t^3=B$$
Then solve for:
$$y=s-t$$
Can you find the roots now?
Once you have these roots, you can perform whatever operations on them you like.

P.S. You can find more information about Cardano's method here.