Mathematical‌ Inquiry‌ about‌ Factorizing‌ the‌ Polynomial‌ of‌ x<sup>n</sup>-1‌ and‌ its‌ Phase

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  • 整式ｘ<sup>n</sup>－１の因数分解に関する数学的探究とその様相
  • 整式xⁿ-1の因数分解に関する数学的探究とその様相
  • セイシキ x ⁿ-1 ノ インスウ ブンカイ ニ カンスル スウガクテキ タンキュウ ト ソノ ヨウソウ

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Abstract

<p>　　 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.</p><p>　　 As a result of our discussions, we obtained several insights:</p><p>(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.</p><p>(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.</p><p>(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.</p>

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