数学 - Python - 代数学 - 因数分解と分数式 - 整式の最大公約数、最小公倍数

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)、練習問題の問2の解答を求めてみる。

1. $\begin{array}{l} x^{4} - y^{4} \\ = (x^{2} + y^{2}) (x^{2} - y^{2}) \\ = (x^{2} + y^{2}) (x + y) (x - y) \\ x^{3} - y^{3} \\ = (x - y) (x^{2} + x y + y^{2}) \end{array}$
  よって、最大公約数は
  $x - y$
  最小公倍数は
  $(x^{2} + x y + y^{2}) (x^{2} + y^{2}) (x + y) (x - y)$
2. $\begin{array}{l} 2 a b - a^{2} - b^{2} + c^{2} \\ = c^{2} - {(a - b)}^{2} \\ = (a - b + c) (- a + b + c) \\ a^{2} - b^{2} + c^{2} + 2 a c \\ = {(a + c)}^{2} - b^{2} \\ = (a + b + c) (a - b + c) \\ a - b + c \\ (a + b + c) (- a + b + c) (a - b + c) \end{array}$
3. $\begin{array}{l} x^{4} - 27 x^{2} y^{2} + y^{4} \\ = {(x^{2} - y^{2})}^{2} - 25 x^{2} y^{2} \\ = {(x^{2} - y^{2})}^{2} - {(5 x y)}^{2} \\ = (x^{2} + 5 x y - y^{2}) (x^{2} - 5 x y - y^{2}) \\ x^{2} + 5 x y - y^{2} \\ x^{4} - 27 x^{2} y^{2} + y^{4} \end{array}$
4. $\begin{array}{l} 4 a^{2} - 9 \\ = (2 a - 3) (2 a + 3) \\ 6 a^{2} + 5 a - 6 \\ = (2 a + 3) (3 a - 2) \\ 6 a^{2} + 13 a + 6 \\ = (2 a + 3) (3 a + 2) \\ 2 a + 3 \\ (2 a - 3) (2 a + 3) (3 a + 2) (3 a - 2) \\ = (4 a^{2} - 9) (9 a^{2} - 4) \\ = 36 a^{4} - 97 a^{2} + 36 \end{array}$

#!/usr/bin/env python3 from unittest import TestCase, main from sympy import symbols, gcd, lcm print('2.') x, y, a, b, c = symbols('x, y, a, b, c', real=True) class MyTest(TestCase): def test1(self): exprs = (x ** 4 - y ** 4, x ** 3 - y ** 3) self.assertEqual(gcd(exprs), x - y) self.assertEqual(lcm(exprs), ((x ** 2 + x * y + y ** 2) * (x ** 2 + y ** 2) * (x + y) * (x - y)).expand()) def test2(self): exprs = (2 * a * b - a ** 2 - b ** 2 + c ** 2, a ** 2 - b ** 2 + c ** 2 + 2 * a * c) self.assertEqual(gcd(exprs), a - b + c) self.assertEqual(lcm(exprs), ((a + b + c) * (- a + b + c) * (a - b + c)).expand()) def test3(self): exprs = (x ** 4 - 27 * x ** 2 * y ** 2 + y ** 4, x ** 2 + 5 * x * y - y ** 2) self.assertEqual(gcd(exprs), x ** 2 + 5 * x * y - y ** 2) self.assertEqual(lcm(exprs), (x ** 4 - 27 * x ** 2 * y ** 2 + y ** 4)) def test4(self): exprs = (4 * a ** 2 - 9, 6 * a ** 2 + 5 * a - 6, 6 * a ** 2 + 13 * a + 6) self.assertEqual(gcd(exprs), 2 * a + 3) self.assertEqual(lcm(exprs), 36 * a ** 4 - 97 * a ** 2 + 36) if __name__ == '__main__': main()

% ./sample2.py -v 2. test1 (__main__.MyTest) ... ok test2 (__main__.MyTest) ... ok test3 (__main__.MyTest) ... ok test4 (__main__.MyTest) ... ok ---------------------------------------------------------------------- Ran 4 tests in 0.103s OK %

Kamimura's blog

2020年1月20日月曜日

数学 - Python - 代数学 - 因数分解と分数式 - 整式の最大公約数、最小公倍数

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