Three-variable system of simultaneous equations 
$x + y + z = 4$
  $x^2 + y^2 + z^2 = 4$
  $x^3 + y^3 + z^3 = 4$  

Any ideas on how to solve for $(x,y,z)$ satisfying the three simultaneous equations, provided there can be both real and complex solutions?

For a fixed number of variables and a fixed power $n$ the sum of powers $$x^n + y^n + z^n + ... + w^n$$ is a symmetric polynomial.
It is expressible in terms of elementary symmetric polynomials. The elementary symmetric polynomials for three variables are


*

*$e_1 = x + y + z$

*$e_2 = x y + x z + y z$

*$e_3 = x y z$


and your polynomials expressed in terms of them are


*

*$x + y + z = e_1$

*$x^2 + y^2 + z^2 = e_1^2 - 2 e_2$

*$x^3 + y^3 + z^3 = e_1^3 - 3(e_1 e_2 - e_3)$


Now we can find the values of $e_1,e_2,e_3$ evaluated at the given $x,y,z$:
$e_1 = 4$, $e_2 = 6$, $e_3 = 4$.
Now consider the polynomial $(t - x)(t - y)(t - z) = t^3 - e_1 t^2 + e_2 t - e_3 = t^3 - 4 t^2 + 6 t - 4$.
It has the solutions $t = 2, 1 + i$ and $1 - i$.
So now we can check if these are correct:


*

*$(2) + (1+i) + (1-i) = 4$

*$(2)^2 + (1+i)^2 + (1-i)^2 = 4 + 2i - 2i = 4$

*$(2)^3 + (1+i)^3 + (1-i)^3 = 8 + -2 + 2i -2 - 2i = 4$

