数学 - Python - 代数学 - 因数分解と分数式 - 分数式とその計算 - 分数式の演算、乗法と除法

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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)、3(分数式とその計算)の問14の解答を求めてみる。

1. $\begin{array}{l} \frac{36 x y^{3}}{5 a^{2}} \cdot \frac{25 a^{2} x}{12 y^{3}} \\ = 15 x^{2} \end{array}$
2. $\frac{2}{3 a b c d}$
3. $\begin{array}{l} \frac{x y^{2} (y^{2} - x^{2})}{3 x^{2} y (x + 3 y)} \cdot \frac{6 x^{2}}{5 y^{2}} \\ = \frac{2}{5} \cdot \frac{x}{y} \cdot \frac{y^{2} - x^{2}}{x + 3 y} \\ = \frac{2 x (y^{2} - x^{2})}{5 y (x + 3 y)} \end{array}$
4. $\begin{array}{l} \frac{(x + 7) (x - 7) (x + 2)}{x (x + 2) (x - 7)} \\ = \frac{x + 7}{x} \end{array}$
5. $\begin{array}{l} \frac{(x + y) {(x - y)}^{2}}{{(x - y)}^{2} x (x + y)} \\ = \frac{1}{x} \end{array}$
6. $\begin{array}{l} \frac{(x + 1) (x + 2) (x + 4) (x - 3)}{(x - 2) (x - 3) (x + 1) (x + 3)} \\ = \frac{(x + 2) (x + 4)}{(x - 2) (x + 3)} \end{array}$
7. $\begin{array}{l} \frac{5 (x - 1) (x + 2) (x^{2} - 2 x + 4)}{(x - 6) (x + 2) 4 (x + 1) (x - 1)} \\ = \frac{5 (x^{2} - 2 x + 4)}{4 (x - 6) (x + 1)} \end{array}$
8. $\begin{array}{l} \frac{(x - 3) (x - 2) 2 (3 x^{2} + 5 x + 2)}{(3 x - 2) (x + 1) (x - 3) (x + 2)} \\ = \frac{2 (x - 3) (x - 2) (3 x + 2) (x + 1)}{(3 x - 2) (x - 3) (x + 1) (x + 2)} \\ = \frac{2 (x - 2) (3 x + 2)}{(3 x - 2) (x + 2)} \end{array}$
9. $\begin{array}{l} \frac{(1 + a) (1 - a) (1 + b) (1 - b)}{(1 + b) a (1 + a) (1 - a)} \\ = \frac{1 - b}{a} \end{array}$
10. $\begin{array}{l} \frac{(2 x - 5) (3 x + 4) {(x - 1)}^{2} x (x + 2)}{(x + 2) (x - 2) 3 x (2 x - 5) (x + 1) (3 x + 4)} \\ = \frac{{(x - 1)}^{2}}{3 (x - 2) (x + 1)} \end{array}$

#!/usr/bin/env python3 from unittest import TestCase, main from sympy import symbols, Rational print('14.') class MyTest(TestCase): def test(self): x, y, a, b, c, d = symbols('x, y, a, b, c, d') spam = [36 * x * y ** 3 * 25 * a ** 2 * x / (5 * a ** 2) / (12 * y ** 3), 8 * c ** 2 * d ** 2 / (9 * a ** 2 * b ** 2) * 3 * a * b / (4 * c ** 3 * d ** 3), (x * y ** 4 - x ** 3 * y ** 2) * 6 * x ** 2 / (3 * x ** 3 * y + 9 * x ** 2 * y ** 2) / (5 * y ** 2), (x ** 2 - 49) * (x + 2) / (x ** 2 + 2 * x) / (x - 7), (x ** 2 - y ** 2) * (x - y) / (x ** 2 - 2 * x * y + y ** 2) / (x ** 2 + x * y), (x ** 2 + 3 * x + 2) * (x ** 2 + x - 12) / (x ** 2 - 5 * x + 6) / (x ** 2 + 4 * x + 3), (5 * x - 5) * (x ** 3 + 8) / (x ** 2 - 4 * x - 12) / (4 * x ** 2 - 4), (x ** 2 - 5 * x + 6) * (6 * x ** 2 + 10 * x + 4) / (3 * x ** 2 - x - 2) / (x ** 2 - x - 6), (1 - a ** 2) * (1 - b ** 2) * 1 / (1 + b) / (a + a ** 2) / (1 - a), (6 * x ** 2 - 7 * x - 20) * (x ** 2 - x - 2) * (x ** 2 + 2 * x) / (x ** 2 - 4) / (6 * x ** 2 - 15 * x) / (3 * x ** 2 + 7 * x + 4)] egg = [15 * x ** 2, 2 / (3 * a * b * c * d), 2 * x * (y ** 2 - x ** 2) / (5 * y * (x + 3 * y)), (x + 7) / x, 1 / x, (x + 2) * (x + 4) / (x - 2) / (x + 3), 5 * (x ** 2 - 2 * x + 4) / 4 / (x - 6) / (x + 1), 2 * (x - 2) * (x + 1) / (x - 1) / (x + 2), (1 - b) / a, Rational(1, 3)] for s, t in zip(spam, egg): self.assertEqual(s.factor(), t.factor()) if __name__ == '__main__': main()

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2020年1月5日日曜日

数学 - Python - 代数学 - 因数分解と分数式 - 分数式とその計算 - 分数式の演算、乗法と除法

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