jmc

The polynomial $x^4+1=0$ shows that $n$ can be 0

The polynomial $x(x^3+2)$ shows that $n$ can be 1.

The polynomial $x^2(x^2+1)$ shows that $n$ can be 2.

The polynomial $x^4$ shows that $n$ can be 4.

Suppose the polynomial has three integer roots. By Vieta's formulas, the sum of the roots is $-b,$ which is an integer. Therefore, the fourth root is also an integer, so it is impossible to have exactly three integer roots.

Thus, the possible values of $n$ are $\boxed{0,1,2,4}.$