Computers and the Rational-Root Theorem

Two recent articles in the Mathematics Teacher have described methods of finding the roots of polynomials by use of the computer (Kimberling 1985a, 1985b). Of the two methods presented, the half interval search is more acceptable to students, since it does not require a knowledge of the calculus. Both techniques approximate roots to a high degree of accuracy, but neither of these techniques supplies exact rational roots. Students who use these methods a re often concerned to find that a root of 0.5 may be approximated as 0.499999 or that a root of 3 is approximated as 2.99999. The question arises as to whether or not these rational roots can be found on the computer and printed as rational numbers. Using a topic that many of my students study in second-year algebra, the rational-root theorem, it is possible to write the desired program. A complete program listing is shown in program 1