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.
+605 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).
