## Polynomial Remainder Theorem

If the Coefficients of the Polynomial

 (1)

are specified to be Integers, then integral 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 Order with integral 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 integral roots of (1) are of the form [factors of ]/[factors of ].