Another polynomial equation

Question

Let $r$ be a root of the polynomial $p(x)=(\sqrt{3}-\sqrt{2})x^3 + \sqrt{2}x-\sqrt{3}+1$. Find another polynomial $q(x)$, with all integer coefficients, such that $q(r)=0$.

Answer 1

$\sqrt{3}r^3-\sqrt{2}r^3+\sqrt{2}r-\sqrt{3}+1=0\\\sqrt{3}r^3-\sqrt{3}+1=\sqrt{2}r^3-\sqrt{2}r$

Square both sides

$3(r^3-1)^2+1+2\sqrt{3}(r^3-1)=2(r^3-r)^2$

$3(r^3-1)^2+1-2(r^3-r)^2=-2\sqrt{3}(r^3-1)$

Square both sides again

$(3(r^3-1)^2+1-2(r^3-r)^2)^2-12(r^3-1)^2=0$

Therefore $q(x)=(3(x^3-1)^2+1-2(x^3-x)^2)^2-12(x^3-1)^2$