The purpose of this paper is to investigate senior high school students’ mathematical inquiry and its phase in factorizing the polynomial xn-1, which focus on producing the irreducible polynomial, discovering and proving mathematical character about the irreducible polynomial xn-1. We investigated students’ mathematical inquiry by teaching units about factorizing the polynomial xn-1 by qualitative methods.
As a result of our discussions, we obtained several insights:
(1) By its behavior and its product about factorizing the polynomial xn-1 inductively, a student’s mathematical inquiry is encouraged to connect the properties of the polynomial xn-1 with the properties of the integer n.
(2) Various interpretations about factorizing the polynomial x6-1 become a foundation of factorizing the polynomial xn-1. For example, students can connect factorizing the polynomial x6-1 with established facts about factorizing when they read the exponential number 6 of x6-1 to 6=2×3 or 6=3×2. Similarly, students advance their own thinking by changing the role of polynomial xn-1 as the base.
(3) Algebraic matters that students want to prove are generated in inquiry about factorizing the polynomial xn-1 with using a Computer Algebra System (CAS). However, various students’ activities about proving these matters exist.