jmc

By the Rational Root Theorem, the only possible rational roots are of the form $\pm \frac{a}{b},$ where $a$ divides 4 and $b$ divides 2. Thus, the possible rational roots are $$\pm \frac{1}{2},\pm 1,\pm 2,\pm 4.$$Thus, there are $\boxed{8}$ possible rational roots.