General form of a factorable polynomial

Abstract

This paper provides extension to the types of factoring of polynomials to illustrate the fact that multiplication with polynomials has no restrictions when it comes to the number of terms so does factoring. It aims to develop students’ perception about polynomials and to motivate them to continue discovering factorable polynomials.that are new to us.The idea originated from the expression x⁴ + x² y² + y⁴ which is a product of x² + xy + y² and x² – xy + y². Factor theorem, rational zeros of a polynomial, and symmetric and transitive properties of real numbers were used in this study in proving other factorable polynomials. The contents include the difference of perfect nth powers (ax)ⁿ – (by)ⁿ, the trinomial of the form   xⁿ + (xy)^n/2 + yⁿ, and the general form of a factorable polynomial p(x) = xⁿ + x^{n – d} y^d + x^{n – 2d} y^2d + … + x^2dy^{n – 2d} + x^dy^{n – d} + yⁿ for some positive integers n and d with a and b as real numbers.