数学 - Python - 代数学 - 因数分解と分数式 - 因数分解 - 展開の逆、公式

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)、1(因数分解)の問2の解答を求めてみる。

1. $\begin{array}{l} 25 x^{2} - 20 x + 4 \\ = {(5 x - 2)}^{2} \end{array}$
2. $\begin{array}{l} \frac{1}{4} x^{2} - \frac{1}{5} x y + \frac{1}{25} y \\ = {(\frac{1}{2} x - \frac{1}{5} y)}^{2} \end{array}$
3. $\begin{array}{l} 4 a^{2} + 12 a + 9 \\ = {(2 a + 3)}^{2} \end{array}$
4. $\begin{array}{l} \frac{1}{4} {(x + y)}^{2} - a (x + y) + a^{2} \\ = {(\frac{1}{2} (x + y) - a)}^{2} \\ = {(\frac{1}{2} x + \frac{1}{2} y - a)}^{2} \\ = \frac{1}{4} {(x + y - 2 a)}^{2} \end{array}$
5. $\begin{array}{l} 36 x^{3} + 4 x^{2} y + \frac{1}{9} x y^{2} \\ = x (36 x^{2} + 4 x y + \frac{1}{9} y^{2}) \\ = x {(6 x + \frac{1}{3} y)}^{2} \\ = \frac{1}{9} x {(18 x + y)}^{2} \end{array}$
6. $\begin{array}{l} 25 a^{2} b^{2} - 9 c^{2} d^{2} \\ = (5 a b - 3 c d) (5 a b + 3 c d) \end{array}$
7. $\begin{array}{l} {(a - 2 b)}^{2} - {(3 a + 4 b)}^{2} \\ = (a - 2 b + 3 a + 4 b) (a - 2 b - 3 a - 4 b) \\ = (4 a + 2 b) (- 2 a - 6 b) \\ = - 4 (2 a + b) (a + 3 b) \end{array}$
8. $\begin{array}{l} x^{2} y^{2} - x^{2} - y^{2} + 1 \\ = (y^{2} - 1) x^{2} - (y^{2} - 1) \\ = (1^{2} - 1) (x^{2} - 1) \\ = (x - 1) (x + 1) (y - 1) (y + 1) \end{array}$
9. $\begin{array}{l} a^{4} - b^{4} \\ = (a^{2} + b^{2}) (a^{2} - b^{2}) \\ = (a^{2} + b^{2}) (a + b) (a - b) \end{array}$
10. $\begin{array}{l} a^{5} - a \\ = a (a^{4} - 1) \\ = a (a^{2} + 1) (a^{2} - 1) \\ = (a^{2} + 1) a (a + 1) (a - 1) \end{array}$
11. $\begin{array}{l} 16 x^{2} - 8 x y + y^{2} - 9 z^{2} \\ = {(4 x - y)}^{2} - 9 z^{2} \\ = (4 x - y - 3 z) (4 x - y + 3 z) \end{array}$
12. $\begin{array}{l} 4 a^{2} b^{2} - {(a^{2} + b^{2} - c^{2})}^{2} \\ = (2 a b - (a^{2} + b^{2} - c^{2})) (2 a b + (a^{2} + b^{2} - c^{2})) \\ = - ({(a - b)}^{2} - c^{2}) ({(a + b)}^{2} - c^{2}) \\ = - (a - b + c) (a - b - c) (a + b + c) (a + b - c) \end{array}$

#!/usr/bin/env python3 from unittest import TestCase, main from sympy import symbols print('2.') class MyTest(TestCase): def test(self): x, y, z, a, b, c, d = symbols('x, y, z, a, b, c, d') spam = [25 * x ** 2 - 20 * x + 4, x ** 2 / 4 - x * y / 5 + y ** 2 / 25, 4 * a ** 2 + 12 * a + 9, (x + y) ** 2 / 4 - a * (x + y) + a ** 2, 36 * x ** 3 + 4 * x ** 2 * y + x * y ** 2 / 9, 25 * a ** 2 * b ** 2 - 9 * c ** 2 * d ** 2, (a - 2 * b) ** 2 - (3 * a + 4 * b) ** 2, x ** 2 * y ** 2 - x ** 2 - y ** 2 + 1, a ** 4 - b ** 4, a ** 5 - a, 16 * x ** 2 - 8 * x * y + y ** 2 - 9 * z ** 2, 4 * a ** 2 * b ** 2 - (a ** 2 + b ** 2 - c ** 2) ** 2] egg = [(5 * x - 2) ** 2, (x / 2 - y / 5) ** 2, (2 * a + 3) ** 2, (x + y - 2 * a) ** 2 / 4, x * (18 * x + y) ** 2 / 9, (5 * a * b - 3 * c * d) * (5 * a * b + 3 * c * d), -4 * (2 * a + b) * (a + 3 * b), (x - 1) * (x + 1) * (y - 1) * (y + 1), (a ** 2 + b ** 2) * (a + b) * (a - b), (a ** 2 + 1) * a * (a + 1) * (a - 1), (4 * x - y - 3 * z) * (4 * x - y + 3 * z), -(a - b + c) * (a - b - c) * (a + b + c) * (a + b - c)] for s, t in zip(spam, egg): self.assertEqual(s.expand(), t.expand()) if __name__ == '__main__': main()

Kamimura's blog

2019年12月21日土曜日

数学 - Python - 代数学 - 因数分解と分数式 - 因数分解 - 展開の逆、公式

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