Number of "possible" rational roots of a univariate polynomial

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Is it possible to determine the number of possible rational roots of a single variable polynomial?

    polynomial a0 + a1x + a2x^2 + ... + anx^n.

This is to find all integers p that is a factor of a0 and for all integers q that is a factor of an, check if the rational number p/q is a root for the input polynomial.

I thought of simply multiplying the number of factors of a0 and an, but wouldn't the possibility of duplicates exist?

I need it to create a program that will get the rational roots. Thanks :)