Rational Zero Theorem

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 denominatorwhich 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 ].

REFERENCES:

Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, p. 34, 1982.

Niven, I. M. Numbers: Rational and Irrational. New York: Random House, 1961.