A polynomial with integer coefficients is of the form
\[2x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 1 = 0.\]Find the number of different possible rational roots of this polynomial.
I know that if a polynomial has a rational root, then it must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. This is called the rational root theorem. In this case, the constant term is 1, so its only factors are 1 and -1. The leading coefficient is 2, so its factors are 1, -1, 2, and -2. Therefore, the possible rational roots are \[\pm 1, \pm \frac{1}{2}.\] There are only four possible rational roots, and they are all different. 