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、加法と減法、通分

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)、3(分数式とその計算)の問13の解答を求めてみる。

1. $\begin{array}{l} \frac{(a - b) c + a (b - c) + b (c - a)}{a b c} \\ = 0 \end{array}$
2. $\begin{array}{l} \frac{1 - x + 2 x}{1 - x^{2}} \\ = \frac{1 + x}{1 - x^{2}} \\ = \frac{1}{1 - x} \end{array}$
3. $\begin{array}{l} \frac{x - 4 - (x + 3)}{x^{2} - 16} \\ = - \frac{7}{x^{2} - 16} \end{array}$
4. $\begin{array}{l} \frac{{(x + 1)}^{2} - (x^{2} + 3)}{2 (x^{2} - 1)} \\ = \frac{x^{2} + 2 x + 1 - x^{2} - 3}{2 (x^{2} - 1)} \\ = \frac{2 (x - 1)}{2 (x^{2} - 1)} \\ = \frac{1}{x + 1} \end{array}$
5. $\begin{array}{l} \frac{x + 8}{(x - 1) (x + 2)} - \frac{x + 4}{(x + 1) (x + 2)} \\ = \frac{(x + 1) (x + 8) - (x - 1) (x + 4)}{(x - 1) (x + 1) (x + 2)} \\ = \frac{x^{2} + 9 x + 8 - x^{2} - 3 x + 4}{(x - 1) (x + 1) (x + 2)} \\ = \frac{6 (x + 2)}{(x - 1) (x + 1) (x + 2)} \\ = \frac{6}{x^{2} - 1} \end{array}$
6. $\begin{array}{l} \frac{{(x + 1)}^{2} - (x - 2) (x + 2) + x^{2} - 4}{(x + 2) (x + 1)} \\ = \frac{x^{2} + 2 x + 1 - x^{2} + 4 + x^{2} - 4}{(x + 1) (x + 2)} \\ = \frac{x^{2} + 2 x + 1}{(x + 1) (x + 2)} \\ = \frac{x + 1}{x + 2} \end{array}$
7. $\begin{array}{l} \frac{(x + 2) (x + 1) - 2 x}{x + 1} - \frac{3 x^{2} + 4}{x (x + 1)} \\ = \frac{x^{2} + x + 2}{x + 1} - \frac{3 x^{2} + 4}{x (x + 1)} \\ = \frac{x^{3} - 2 x^{2} + 2 x - 4}{x (x + 1)} \end{array}$
8. $\begin{array}{l} \frac{x^{2} - x - 2 - x^{2} + x - 1}{x^{3} + 1} + \frac{x^{2} + x + 3}{x^{3} + 1} \\ = \frac{x^{2} + x}{x^{3} + 1} \end{array}$
9. $\begin{array}{l} \frac{2 a}{a^{2} - 1} + \frac{2 a}{a^{2} + 1} + \frac{4 a^{3}}{a^{4} + 1} \\ = \frac{4 a^{3}}{a^{4} - 1} + \frac{4 a^{3}}{a^{4} + 1} \\ = \frac{8 a^{7}}{a^{8} - 1} \end{array}$
10. $\begin{array}{l} \frac{x^{3} + x + x}{{(x^{2} + 1)}^{2}} + \frac{1}{x {(x^{2} + 1)}^{2}} \\ = \frac{x^{4} + 2 x^{2} + 1}{x {(x^{2} + 1)}^{2}} \\ = \frac{{(x^{2} + 1)}^{2}}{x {(x^{2} + 1)}^{2}} \\ = \frac{1}{x} \end{array}$
11. $\begin{array}{l} \frac{x - 2}{(2 x - 3) (x - 1)} + \frac{3 x - 1}{(2 x - 3) (x + 2)} + \frac{2 x^{2} - 5}{(x + 2) (x - 1)} \\ = \frac{x^{2} - 4 + 3 x^{2} - 4 x + 1}{(2 x - 3) (x - 1) (x + 2)} + \frac{2 x^{2} - 5}{(x + 2) (x - 1)} \\ = \frac{4 x^{2} - 4 x - 3}{(2 x - 3) (x - 1) (x + 2)} + \frac{4 x^{3} - 6 x^{2} - 10 x + 15}{(2 x - 3) (x - 1) (x + 2)} \\ = \frac{4 x^{3} - 2 x^{2} - 14 x + 12}{(2 x - 3) (x - 1) (x + 2)} \end{array}$

#!/usr/bin/env python3 from unittest import TestCase, main from sympy import symbols print('13.') class MyTest(TestCase): def test(self): x, a, b, c = symbols('x, a, b, c') spam = [(a - b) / (a * b) + (b - c) / (b * c) + (c - a) / (c * a), 1 / (1 + x) + 2 * x / (1 - x ** 2), 1 / (x + 4) - (x + 3) / (x ** 2 - 16), (x + 1) / (2 * x - 2) - (x ** 2 + 3) / (2 * x ** 2 - 2), (x + 8) / (x ** 2 + x - 2) - (x + 4) / (x ** 2 + 3 * x + 2), (x + 1) / (x + 2) - (x - 2) / (x + 1) + (x ** 2 - 4) / (x ** 2 + 3 * x + 2), x + 2 - 2 * x / (x + 1) - (3 * x ** 2 + 4) / x / (x + 1), (x - 2) / (x ** 2 - x + 1) - 1 / (x + 1) + (x ** 2 + x + 3) / (x ** 3 + 1), 1 / (a - 1) + 1 / (a + 1) + 2 * a / (a ** 2 + 1) + 4 * a ** 3 / (a ** 4 + 1), x / (x ** 2 + 1) + x / (x ** 2 + 1) ** 2 + 1 / (x * (x ** 2 + 1) ** 2), (x - 2) / (2 * x ** 2 - 5 * x + 3) + (3 * x - 1) / (2 * x ** 2 + x - 6) + (2 * x ** 2 - 5) / (x ** 2 + x - 2), a / ((a - b) * (a - c)) + b / (b - c) / (b - a) + c / (c - a) / (c - b)] egg = [0, 1 / (1 - x), -7 / (x ** 2 - 16), 1 / (x + 1), 6 / (x ** 2 - 1), (x + 1) / (x + 2), (x ** 3 - 2 * x ** 2 + 2 * x - 4) / x / (x + 1), (x ** 2 + x) / (x ** 3 + 1), 8 * a ** 7 / (a ** 8 - 1), 1 / x, (4 * x ** 3 - 2 * x ** 2 - 14 * x + 12) / (2 * x - 3) / (x - 1) / (x + 2)] for s, t in zip(spam, egg): self.assertEqual((s - t).simplify(), 0) if __name__ == '__main__': main()

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2020年1月4日土曜日

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、加法と減法、通分

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