Solving $x+y+z+u+t=0$, $x^3+y^3+z^3+u^3+t^3=0$, $x^5+y^5+z^5+u^5+t^5=-10$

Question

Find all real numbers $x$, $y$, $z$, $t$, $u$ that

$x+y+z+u+t=0$

$x^3+y^3+z^3+u^3+t^3=0$

$x^5+y^5+z^5+u^5+t^5=-10$

I'm learning about Chebyshev polynominals but in this case, I still haven't got any idea :(( please help me

Answer 1

There are many real solutions. To obtain examples, we choose, say, $x=-2$ and $y=3$. Taking resultants we obtain the equation $$ 3u^3+3u^2-19u-1=0, $$ and $z$ as a root of a quadratic equation in $u$. Now there are three real solutions, e.g., $u=2.09659797695$. Then $$ (x,y,z,u,t)=(-2,3,- 3.04437453374,2.09659797695,- 0.0522234432082) $$ is a solution.