contest math - Find a polynomial with integer coefficients

Find a polynomial $p$ with integer coefficients for which $a = \sqrt{2} + \sqrt[3]{2}$ is a root. That is find $p$ such that for some non-negative integer $n$, and integers $a_0$, $a_1$, $a_2$, ..., $a_n$, $p(x) = a_0 + a_1 x + a_2 x^2 + ... + a_n x^n$, and $p(a) = 0$.

I do not know how to solve this. It is very challenging. Also, if you name any theorem please describe it in a way that is easy to understand. If you just name it, I won't be able to understand it. (My math might not be/is not as good as yours.)

Thanks for any help!

Answer

We have
$$

a=\sqrt2+\sqrt[3]2\\
a-\sqrt2=\sqrt[3]2\\
(a-\sqrt2)^3=2\\
a^3-3\sqrt2a^2+6a-2\sqrt2=2\\
a^3+6a-2=\sqrt2(3a^2+2)\\
(a^3+6a-2)^2=2(3a^2+2)^2
$$
which has all integer coefficients, once you expand the brackets.