Completing nth powers of polynomials

Abstract

A frequent exercise in high school algebra courses is completing the square of some given polynomial. The goal is to find terms involving only constants independent of the main variable, which when added to the given polynomial will result in a perfect square. As a typical example, (x2 + 4x + 3) + 1 &equil; (x+2)2. The method for completing the square such as this one is often nothing more than applying the pattern matching abilities of students to the problem knowing the pattern (x+y)2 &equil; x2 + 2xy + y2. Here, we ask the question whether this problem can be generalized and whether there exists a constructive algorithm that replaces and extends the simple completion procedure of our high school days. The answer turns out to lie in the familiar process of computing polynomial remainder sequences (PRS) [Brown71].