<s>[INST] $x$ is an irrational number such that $x^2 - 2x$ and $x^3 -5x$ are rational. 
If $x$ is an irrational number such that $x^2 - 2x$ and $x^3 -5x$ are rational numbers. What does $x^3 - 5x$ equals to?

 [/INST] By hypothesis there exist $a$ and $b$ such that $x$ is a root of the polynomials $f(X)=X^2-2X+a$ and $g(x)=X^3-5X+b$. Since $x$ is irrational, $[\mathbb Q(x):\mathbb Q]=2$ and the minimal polynomial of $x$ is $f(X)$. This implies that $g(X)$ is a multiple of $f(X)$, so
$$g(X)=(X-c)f(X)$$
and equalising the coefficients we find that $a=-1$, $b=-2$ and $c=-2$. Hence $x$ is a root of $X^2-2X-1$, and $x^3-5x=2$.
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