# 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 $$z_1 + z_2 + z_3 = 0 , z_1 \cdot z_2 + z_1 \cdot z_3 + z_2 \cdot z_3 = -1 , z_1z_2z_3 = 1$$

• Have you tried Cardano's method for solving a depressed cubic? – Micah Windsor Apr 9 at 17:31
• No. What is it? – Yehonatan Apr 9 at 17:33
• I will post an answer. It is quite simple. – Micah Windsor Apr 9 at 17:36
• Thank you Micha! – Yehonatan Apr 9 at 17:37

Here is a simple slightly tedious method:

Let $$A=\begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$$ and note that $$\chi_A(z) = z^3-z-1$$. Hence the sum of the $$5$$th powers of the roots is given by $$\operatorname{tr} A^5 = 5$$.

• Rabbits out of a hat. – Yves Daoust Apr 9 at 17:35
• That matrix is my companion :-). – copper.hat Apr 9 at 17:36
• I didn't learn matrix but thanks. – Yehonatan Apr 9 at 17:41
• @YvesDaoust's method is nice. – copper.hat Apr 9 at 17:42
• Excellent (+1). – Gibbs Apr 9 at 17:45

Hint:

For all roots, $$z^5=z^2z^3=z^3+z^2=z^2+z+1.$$

Expand $$(z_1+z_2+z_3)^2.$$

• @Yehonatan: what is $z^3$ ? – Yves Daoust Apr 9 at 17:37
• I didn't tell you i tell cooper – Yehonatan Apr 9 at 17:38
• And what do you mean at what is $z^3$ ? – Yehonatan Apr 9 at 17:39
• @Yehonatan: observe your problem. – Yves Daoust Apr 9 at 17:39
• +1: Nice observation. – copper.hat Apr 9 at 17:40

$$z_1^3=z_1+1,z_2^3=z_2+1$$

$$z_1z_2z_3 = 1\rightarrow z_1z_2(z_1+z_2)=-1$$

$$z_1^5 + z_2^5 + z_3^5=z_1^5 + z_2^5 - (z_1+z_2)^5= -(5z_1^4z_2+10z_1^3z_2^2+10z_1^2z_2^3+5z_1z_2^4)=-5z_1z_2(z_1^3+2z_1^2z_2+2z_1z_2^2+z_2^3)=-5z_1z_2(z_1+z_2+2+2z_1z_2(z_1+z_2))=-5z_1z_2(z_1+z_2+2-2)=-5z_1z_2(z_1+z_2)= 5$$

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.