If the coefficients of the polynomial
(1)

are specified to be integers, then rational roots must have a numerator which is a factor of and a denominator which is a factor of (with either sign possible). This follows since a polynomial of polynomial order with rational roots can be expressed as
(2)

where the roots are , , ..., and . Factoring out the s,
(3)

Now, multiplying through,
(4)

where we have not bothered with the other terms. Since the first and last coefficients are and , all the rational roots of equation (1) are of the form [factors of ]/[factors of ].