数学 - Python - 代数学 - 因数分解と分数式 - 整式の最大公約数と最小公倍数 - 数係数の省略

学習環境

代数への出発 (新装版数学入門シリーズ) (松坂和夫(著)、岩波書店)の第3章(因数分解と分数式)、2(整式の最大公約数と最小公倍数)の問10の解答を求めてみる。

1. 最大公約数、最小公倍数はそれぞれ、
  $\begin{array}{l} x y^{2} z \\ x^{3} y^{2} z^{3} \end{array}$
  （数係数は省略。以下同様。）
2. $\begin{array}{l} x y \\ x^{4} y^{3} \end{array}$
3. $\begin{array}{l} a^{2} b c \\ a^{3} b^{3} c^{2} \end{array}$
4. $\begin{array}{l} a^{2} (x - 1) \\ a^{3} {(x - 1)}^{3} \end{array}$
5. $\begin{array}{l} (x + 1) (x + 2) \\ x (x + 1) (x + 2) (x + 3) \end{array}$
6. $\begin{array}{l} x^{2} \\ x^{3} (x + 1) {(x + 2)}^{2} {(x + 3)}^{2} \end{array}$
7. $\begin{array}{l} x^{3} - 3 x + 2 \\ = (x - 1) (x - 2) \\ x^{2} - 9 x + 14 \\ = (x - 2) (x - 7) \\ x - 2 \\ (x - 1) (x - 2) (x - 7) \end{array}$
8. $\begin{array}{l} x^{2} - 9 \\ = (x + 3) (x - 3) \\ x^{2} - 2 x - 3 \\ = (x - 3) (x + 1) \\ x^{2} + 4 x + 3 \\ = (x + 1) (x + 3) \\ 1 \\ (x - 3) (x + 1) (x + 3) \end{array}$
9. $\begin{array}{l} x^{2} + x - 6 \\ = (x - 2) (x + 3) \\ x^{2} - 6 x + 8 \\ = (x - 4) (x - 2) \\ x^{2} - x - 12 \\ = (x - 4) (x + 3) \\ 1 \\ (x - 4) (x - 2) (x + 3) \end{array}$
10. $\begin{array}{l} a^{2} + 2 a b + b^{2} \\ = {(a + b)}^{2} \\ a^{2} - b^{2} \\ = (a - b) (a + b) \\ a^{3} + b^{3} \\ = (a + b) (a^{2} - a b + b^{2}) \\ a + b \\ (a - b) {(a + b)}^{2} (a^{2} - a b + b^{2}) \end{array}$
11. $\begin{array}{l} x^{2} + 5 x + 6 \\ = (x + 2) (x + 6) \\ x^{2} + 6 x + 9 \\ = {(x + 3)}^{2} \\ x^{3} + 8 \\ = (x + 2) (x^{2} - 2 x + 4) \\ 1 \\ (x + 2) {(x + 3)}^{2} (x + 6) (x^{2} - 2 x + 4) \end{array}$
12. $\begin{array}{l} x^{2} + 2 x - 3 \\ = (x - 1) (x + 3) \\ x^{3} + 3 x^{2} - x - 3 \\ = (x - 1) (x^{2} + 4 x + 3) \\ = (x - 1) (x + 1) (x + 3) \\ x^{3} + 2 x^{2} - x - 2 \\ = (x - 1) (x^{2} + 3 x + 2) \\ = (x - 1) (x + 1) (x + 2) \\ x - 1 \\ (x - 1) (x + 1) (x + 2) (x + 3) \end{array}$

#!/usr/bin/env python3 from unittest import TestCase, main from sympy import symbols, gcd, lcm print('9.') class MyTest(TestCase): def test1(self): x, y, z, a, b, c = symbols('x, y, z, a, b, c', real=True) spam = [(x * y ** 2 * z ** 3, x ** 3 * y ** 2 * z), (x ** 2 * y ** 3, x * y ** 2, x ** 4 * y), (a ** 2 * b ** 3, a ** 3 * b * c ** 2, a ** 2 * b ** 2 * c), (a ** 3 * (x - 1), a ** 2 * (x - 1) ** 3), (x * (x + 1) * (x + 2), (x + 1) * (x + 2) * (x + 3)), (x ** 3 * (x + 2) ** 2 * (x + 1), x ** 2 * (x + 3) * (x + 2), x ** 2 * (x + 1) * (x + 3) ** 2), (x ** 2 - 3 * x + 2, x ** 2 - 9 * x + 14), (x ** 2 - 9, x ** 2 - 2 * x - 3, x ** 2 + 4 * x + 3), (x ** 2 + x - 6, x ** 2 - 6 * x + 8, x ** 2 - x - 12), (a ** 2 + 2 * a * b + b ** 2, a ** 2 - b ** 2, a ** 3 + b ** 3), (x ** 2 + 5 * x + 6, x ** 2 + 6 * x + 9, x ** 3 + 8), (x ** 2 + 2 * x - 3, x ** 3 + 3 * x ** 2 - x - 3, x ** 3 + 2 * x ** 2 - x - 2)] gcds = [x * y ** 2 * z, x * y, a ** 2 * b, a ** 2 * (x - 1), (x + 1) * (x + 2), x ** 2, x - 2, 1, 1, a + b, 1, x - 1] lcms = [x ** 3 * y ** 2 * z ** 3, x ** 4 * y ** 3, a ** 3 * b ** 3 * c ** 2, a ** 3 * (x - 1) ** 3, x * (x + 1) * (x + 2) * (x + 3), x ** 3 * (x + 1) * (x + 2) ** 2 * (x + 3) ** 2, (x - 1) * (x - 2) * (x - 7), (x - 3) * (x + 1) * (x + 3), (x - 4) * (x - 2) * (x + 3), (a - b) * (a + b) ** 2 * (a ** 2 - a * b + b ** 2), (x + 2) * (x + 3) ** 2 * (x ** 2 - 2 * x + 4), (x - 1) * (x + 1) * (x + 2) * (x + 3)] for s, g, l in zip(spam, gcds, lcms): self.assertEqual(gcd(s).factor(), g) self.assertEqual(lcm(s).factor(), l) if __name__ == '__main__': main()

Kamimura's blog

2020年1月1日水曜日

数学 - Python - 代数学 - 因数分解と分数式 - 整式の最大公約数と最小公倍数 - 数係数の省略

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