数学 - Python - 代数学 - 連立方程式と高次方程式 - 高次方程式と因数定理 - 4次方程式の解、因数分解、実数、複素数

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5章(連立方程式と高次方程式)、4(高次方程式と因数定理)、問23の解答を求めてみる。

1. $\begin{array}{l} x^{4} = 1 \\ x^{4} - 1 = 0 \\ (x^{2} + 1) (x^{2} - 1) = 0 \\ x = \pm 1, \pm i \end{array}$
2. $\begin{array}{l} x^{4} + 1 = 0 \\ {(x^{2} + 1)}^{2} - 2 x^{2} = 0 \\ (x^{2} + 1 - \sqrt{2} x) (x^{2} + 1 + \sqrt{2} x) = 0 \\ x^{2} - \sqrt{2} x + 1 = 0 \\ x^{2} + \sqrt{2} x + 1 = 0 \\ x = \frac{\pm \sqrt{2} \pm \sqrt{2 - 4}}{2} \\ = \frac{\pm \sqrt{2} \pm \sqrt{2} i}{2} \\ = \frac{\pm 1 \pm i}{\sqrt{2}} \end{array}$
  
  （複号任意）
3. $\begin{array}{l} x^{4} - 5 x^{2} + 6 = 0 \\ (x^{2} - 2) (x^{2} - 3) = 0 \\ x = \pm \sqrt{2}, \pm \sqrt{3} \end{array}$

#!/usr/bin/env python3 from unittest import TestCase, main from sympy import symbols, I, sqrt, solveset, plot print('23.') x = symbols('x') class TestQuarticEquation(TestCase): def test1(self): self.assertEqual(solveset(x ** 4 - 1), {s * o for s in [-1, 1] for o in [1, I]}) def test2(self): self.assertEqual(solveset(x ** 4 + 1), {((s + t * I) / sqrt(2)).expand() for s in [-1, 1] for t in [-1, 1]}) def test3(self): self.assertEqual(solveset(x ** 4 - 5 * x ** 2 + 6), {s * sqrt(o) for s in [-1, 1] for o in [2, 3]}) p = plot(x ** 4, 1, -1, x ** 4 - 5 * x ** 2 + 6, (x, -5, 5), ylim=(-5, 5), legend=True, show=False) colors = ['red', 'green', 'blue', 'brown', 'orange', 'pink'] for i, s in enumerate(p): s.line_color = colors[i] p.show() p.save('sample23.png') if __name__ == "__main__": main()

% ./sample23.py -v 23. test1 (__main__.TestQuarticEquation) ... ok test2 (__main__.TestQuarticEquation) ... ok test3 (__main__.TestQuarticEquation) ... ok ---------------------------------------------------------------------- Ran 3 tests in 0.093s OK %

Kamimura's blog

2020年4月29日水曜日

数学 - Python - 代数学 - 連立方程式と高次方程式 - 高次方程式と因数定理 - 4次方程式の解、因数分解、実数、複素数

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