Does $x_1^3+x_2^3+\dots+x_n^3+x_1+x_2+\dots+x_n=0$ imply $x_1+x_2+\dots+x_n=0$?

Question

If $x_1,x_2,\dots,x_n \in \mathbb{R}$ can we claim that:

$$x_1^3+x_2^3+\dots+x_n^3+x_1+x_2+\dots+x_n=0$$

implies

$$x_1+x_2+\dots+x_n=0$$ ?

Answer 1

No, we cannot. Here is a counterexample: $n=6$, $x_1=x_2=x_3=x_4=x_5=1$ and $x_6=-2$.