A polynomial with integer coefficients is of the form
\[7x^4 + a_3 x^3 + a_2 x^2 + a_1 x - 14 = 0.\]Find the number of different possible rational roots of this polynomial.
By the Rational Root Theorem, the only possible rational roots are of the form $\pm \frac{a}{b},$ where $a$ divides 14 and $b$ divides 7.  Thus, the possible rational roots are
\[\pm 1, \ \pm 2, \ \pm 7, \ \pm 14, \ \pm \frac{1}{7}, \ \pm \frac{2}{7}.\]Thus, there are $\boxed{12}$ possible rational roots.