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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.

 May 26, 2021
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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).

 May 27, 2021

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