An Exploration of Synthetic Division for Non-Linear Polynomial Divisors

Abstract

As a goal of developing alternative algorithm on division of polynomials whose dividend is 𝑎1𝑥 𝑛 + 𝑎2𝑥 𝑛−1 + 𝑎3𝑥 𝑛−2 + ⋯ + 𝑎𝑛𝑥 + 𝑎𝑛+1 and the divisor is 𝑏1𝑥 𝑚 + 𝑏2𝑥 𝑚−1 + 𝑏3𝑥 𝑚−2 … + 𝑏𝑚𝑥 + 𝑏𝑚+1, where 𝑛 > 𝑚, 𝑎1 ≠ 0, 𝑏1 ≠ 0, and 𝑎𝑖 ′𝑠and 𝑏𝑖 ′𝑠including 𝑎𝑛+1 and 𝑏𝑚+1 are constant, the researcher explored the synthetic division in compact form.The researcher believes that the algorithm in this study is a good alternative in dividing polynomials of higher degrees. In the said compact form, division of only some distinct pairs of non-linear polynomials were presented and served as reference problems for the researcher during the initial exploration. The Division Algorithm for Polynomials theorem was applied in exploring the problems with quadratic, cubic and quartic divisors. This resulted to the development of formulas for the coefficients 𝑡1of the quotient 𝑄(𝑥) and coefficients 𝑟1 of the remainder 𝑅(𝑥) which were considered important parts of the algorithm. Aside from the condition for the inapplicability for non-linear divisors, additional conditions were provided to the problems where the usual synthetic division is inappropriate. At the end, the algorithm on division of polynomials of higher degrees by non-linear divisors was developed using basic research. Illustrative examples of dividing polynomials using the developed algorithms with monic (𝑏1 = 1) and non-monic (𝑏1 ≠ 1) divisors were provided. Results were verified through other existing methods: long division and synthetic division in compact form.