Problem with rational numbers

Question

Let $x\in\mathbb{R}$. Demonstrate that if the numbers $a = x^3–x$ and $b = x^2 +1$ are rational, then $x$ is rational.

Answer 1

Hint: $x^3-x=x\bigl((x^2+1)-2\bigr)$

Answer 2

$x^2+1\in \mathbb Q \implies x^2\in \mathbb Q$

From this, we deduce that $x^3=rx$ for some $r\in \mathbb Q$. It is easy to see that $r=1\implies x=0,\pm 1$ which are all rational. Assume, then, that $r\neq 1$.

But this implies that $x^3-x=x(r-1)\in \mathbb Q\implies x\in \mathbb Q$ as desired.