Find another polynomial with integer coefficients with the same root as this one: $(\sqrt3-\sqrt2)x^3 + \sqrt2x -\sqrt3 + 1$

Question

Basically it says given that s is a root of this polynomial: $(\sqrt3-\sqrt2)x^3 + \sqrt2x -\sqrt3 + 1$, find another polynomial with integer coefficients that has the same root s as well. I'm super stuck and am unsure on how to approach this problem. I attempted to square some stuff but it didn't really work out.

Answer 1

Why, let's just move all $\sqrt3$'s to the right: $$\sqrt2(x-x^3) + 1 = \sqrt3(x^3+1)$$ Now let's square it. $$2(x-x^3)^2 + 2\sqrt2(x-x^3) + 1 = 3(x^3+1)^2$$ Now move all $\sqrt2$'s to one side. $$ 2\sqrt2(x-x^3) = 3(x^3+1)^2-2(x-x^3)^2-1$$ Now square it again, and you'll have your polynomial with integer coefficients.

Answer 2

Let $P_0(x)$ be your polynomial, and $P_1(x)$, $P_2(x)$, $P_3(x)$ the polynomials obtained by

1. substituting $-\sqrt{2}$ for $\sqrt{2}$,
2. substituting $-\sqrt{3}$ for $\sqrt{3}$,
3. doing both substitutions.

Then the product of these, when expanded out, is a polynomial with integer coefficients.