<s>[INST] Determining whether there are solutions to the cubic polynomial equation $x^3 - x = k - k^3$ other than $x = -k$ for a given parameter $k$ Let $k$ be a real parameter, and consider the equation

$$x^3 - x = k - k^3 .$$

Obviously, $x=-k$ is a solution. Is it the only one? How to prove it?
 [/INST] Write it as
$$
x^3+k^3=x+k
$$
so that you get
$$
(x+k)(x^2-kx+k^2)=x+k
$$
If $x=-k$, you have a solution, otherwise
$$
x^2-kx+k^2-1=0
$$
and the roots are
$$
\frac{k\pm\sqrt{4-3k^2}}{2}
$$
so you two other roots when $4-3k^2>0$, just one more if $4-3k^2=0$.
Well, in the case $4-3k^2$ one of the roots could be the same as $-k$. This happens precisely when $-k$ is a root of $x^2-kx+k^2-1=0$, that is, when
$$
k^2+k^2+k^2-1=0
$$
that is, $k=1/\sqrt{3}$ or $k=-1/\sqrt{3}$.
So we have


*

*three solutions for $-2/\sqrt{3}<k<2/\sqrt{3}$, but $k\ne1/\sqrt{3}$ and $k\ne-1/\sqrt{3}$;

*two solutions for $k=\pm2/\sqrt{3}$ or $k=\pm1/\sqrt{3}$;

*one solution for $|k|>2/\sqrt{3}$.
</s>