A polynomial with integer coefficients is of the form
\[9x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 27 = 0.\]
Find the number of different possible rational roots of this polynomial.
From the rational root theorem, the denominator of the root divides 27 and the numerator divides 9. Keeping in mind that negatives are allowed, the answer is $(2d(9))(2d(27))$ where $d(x)$ denotes the number of divisors of $x$ (which is really easy to manually calculate).