Solving vandermonde like systems of equation

Question

$x + y + z = 0$;

$x^2 + y^2 + z^2 = 1$;

$x^3 + y^3 + z^3 = 0$;

I understand that there are multiple solutions which are the permutations of $(\sqrt{ 2 }/2, 0, -\sqrt{2}/2).$

How do i go about solving for it? I have tried the normal brute force gaussian elimination method, Cramer's rule and i still cant get the answer.

Would appreciate if someone could provide me with an algorithm and/or the steps.

Thank you very much!!

Answer 1

This is easy to solve by a direct calculation. Not even Vieta is needed. Substituting $z=-x-y$ the last equation gives $$ xy(x+y)=0 $$ Obviously $x=y=0$ gives a contradiction, and also we may assume $xy\neq 0$; for $y=-x$ then the second equation is now very easy.

Answer 2

We can use the Viete's theorem.

Indeed, $$0=(x+y+z)^2=1+2(xy+xz+yz),$$ which gives $$xy+xz+yz=-\frac{1}{2}.$$ Also, since $$(x+y+z)^3=x^3+y^3+z^3+3(xy+xz+yz)(x+y+z)-3xyz,$$ we obtain $$xyz=0,$$ which gives that $x$, $y$ and $z$ are roots of the equation: $$t^3-\frac{1}{2}t=0.$$