Finding $x^8+y^8+z^8$, given $x+y+z=0$, $\;xy +xz+yz=-1$, $\;xyz=-1$ 
The system says
$$x+y+z=0$$
$$xy +xz+yz=-1$$
$$xyz=-1$$
Find
$$x^8+y^8+z^8$$

With the first equation I squared and I found that $$x^2+y^2+z^2 =2$$
trying with
$$(x + y + z)^3 = 
 x^3 + y^3 + z^3 +  3 x^2 y + 3 x y^2 + 3 x^2 z++ 3 y^2 z + 3 x z^2 + 
  3 y z^2 + 6 x y z$$
taking advantage of the fact that there is an $xyz=-1$ in the equation, but I'm not getting anywhere, someone less myopic than me.how to solve it?
Thanks
Edit : Will there be any university way to solve this problem , they posed it to a high school friend and told him it was just manipulations of remarkable products.
His answers I understand to some extent but I don't think my friend understands all of it.

Denoting with $e_k(x,y,z), k\geq 0$ the elementary symmetric polynomials
\begin{align*}
e_1(x,y,z)&=x+y+z=0\\
e_2(x,y,z)&=xy+xz+yz=-1\tag{1}\\
e_3(x,y,z)&=xyz=-1
\end{align*}
and with $p_k(x,y,z), k\geq 0$ the $k$-th power sum
\begin{align*}
p_k(x,y,z)=x^k+y^k+z^k\tag{2}
\end{align*}
we recall Newtons identities
admit a generating function representation of the power sums as
\begin{align*}
\color{blue}{\sum_{k=1}^\infty(-1)^{k-1}p_k\frac{t^k}{k}=\ln(1+e_1t+e_2t^2+e_3t^3+\cdots)}\tag{3}
\end{align*}

We obtain from (1) - (3)
\begin{align*}
\color{blue}{\left.x^8+y^8+z^8\right|_{{{e_1=0\ \ }\atop{e_2=-1}}\atop{e_3=-1}}}
&=(-8)[t^8]\ln\left(1-t^2-t^3\right)\\
&=(-8)[t^8]\ln\left(1-t^2(1+t)\right)\\
&=8[t^8]\sum_{j=1}^\infty\frac{t^{2j}(1+t)^j}{j}\tag{4.1}\\
&=8\sum_{j=1}^{4}\frac{1}{j}[t^{8-2j}](1+t)^j\tag{4.2}\\
&=8\sum_{j=1}^4\frac{1}{j}\binom{j}{8-2j}\tag{4.3}\\
&=8\left(\frac{1}{3}\binom{3}{2}+\frac{1}{4}\binom{4}{0}\right)\tag{4.4}\\
&\,\,\color{blue}{=10}
\end{align*}

Comment:

*

*In (4.1) we expand the logarithmic series.


*In (4.2) we apply the rule $[t^{p-q}]A(t)=[t^p]t^qA(t)$. We also set the upper limit of the series to $4$ since other terms do not contribute.


*In (4.3) we select the coefficient of $t^{8-2j}$.


*In (4.4) we use $\binom{p}{q}=0$ if $0<p<q$ are positive integers.
