EJMT Abstract

Synthetic division, as developed by Ruffini in 1804, was limited to division of a polynomial by a linear polynomial factor in the form x - c. Connected to Ruffini’s method, in the early 1800’s, Horner developed techniques for finding roots and determining the derivatives of polynomials. Additionally, Horner expanded Ruffini’s method of synthetic division so that a polynomial could be divided by polynomials of higher degree than just 1. Some high school and college students have used synthetic division to divide by a linear polynomial factor. However, few students may know why and how this process works. The increasing use of computers and calculators with algebraic operating systems may further hide the beauty of synthetic division. This paper: (A) describes how synthetic division can be used in the contexts of integer, rational, real, and complex divisors; (B) investigates Horner’s method of synthetic division to divide by polynomials of any degree; and (C) makes connections with the Remainder Theorem and the Zero Product Property. This paper provides the reader with students investigations and an applet for performing polynomial division.