数学 - Python - 代数学 - 1次方程式, 2次方程式 - 解の係数の関係、2次式の因数分解 - 複素数の範囲

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4章(1次方程式, 2次方程式 )、5(解の係数の関係、2次式の因数分解)、問28の解答を求めてみる。

1. $\begin{array}{l} 56 x^{2} + 89 x + 35 \\ = (7 x + 5) (8 x + 7) \end{array}$
2. $\begin{array}{l} \frac{D}{4} = 1 + 12 = 13 \\ x = \frac{- 1 \pm \sqrt{13}}{3} \\ 3 x^{2} + 2 x - 4 \\ = 3 (x - \frac{- 1 - \sqrt{13}}{3}) (x - \frac{- 1 + \sqrt{13}}{3}) \end{array}$
3. $(2 x - 7) (12 x + 5)$
4. $\begin{array}{l} x = \frac{2 \pm \sqrt{10}}{3} \\ 3 (x - \frac{2 - \sqrt{10}}{3}) (x - \frac{2 + \sqrt{10}}{3}) \end{array}$
5. $\begin{array}{l} 9 x^{2} + 25 \\ = (3 x - 5 i) (3 x + 5 i) \end{array}$
6. $\begin{array}{l} x \\ = \frac{1 \pm \sqrt{1 - 4}}{2} \\ = \frac{1 \pm \sqrt{3} i}{2} \\ (x - \frac{1 - \sqrt{3} i}{2}) (x - \frac{1 + \sqrt{3} i}{2}) \end{array}$
7. $\begin{array}{l} x \\ = \frac{7 \sqrt{2} \pm \sqrt{98 - 26}}{2} \\ = \frac{7 \sqrt{2} \pm \sqrt{72}}{2} \\ = \frac{7 \sqrt{2} \pm 6 \sqrt{2}}{2} \\ = \frac{\sqrt{2}}{2}, \frac{13 \sqrt{2}}{2} \\ 2 (x - \frac{\sqrt{2}}{2}) (x - \frac{13 \sqrt{2}}{2}) \end{array}$
8. $\begin{array}{l} x \\ = \frac{\sqrt{11} \pm \sqrt{11 - 4 \cdot 5}}{2 \sqrt{5}} \\ = \frac{\sqrt{11} \pm 3 i}{2 \sqrt{5}} \\ = \frac{\sqrt{55} \pm 3 \sqrt{5} i}{10} \\ \sqrt{5} (x - \frac{\sqrt{55} - 3 \sqrt{5} i}{10}) (x - \frac{\sqrt{55} + 3 \sqrt{5} i}{10}) \end{array}$

#!/usr/bin/env python3 from unittest import TestCase, main from sympy import symbols, sqrt, plot, I, pprint print('28.') x = symbols('x', image=True) abcs = [(56, 89, 35), (3, 2, -4), (24, -74, -35), (3, -4, -2), (9, 0, 25), (1, -1, 1), (2, -14 * sqrt(2), 13), (sqrt(5), -sqrt(11), sqrt(5))] fs = [a * x ** 2 + b * x + c for a, b, c in abcs] gs = [(7 * x + 5) * (8 * x + 7), 3 * (x - (-1-sqrt(13)) / 3) * (x - (-1 + sqrt(13)) / 3), (2 * x - 7) * (12 * x + 5), 3 * (x - (2 - sqrt(10)) / 3) * (x - (2 + sqrt(10)) / 3), (3 * x - 5 * I) * (3 * x + 5 * I), (x - (1 - sqrt(3) * I) / 2) * (x - (1 + sqrt(3) * I) / 2), 2 * (x - sqrt(2) / 2) * (x - 13 * sqrt(2) / 2), sqrt(5) * (x - (sqrt(55) - 3 * sqrt(5) * I) / 10) * (x - (sqrt(55) + 3 * sqrt(5) * I) / 10) ] class MyTestCase(TestCase): def test(self): for i, (f, g) in enumerate(zip(fs, gs), 1): print(f'({i})') self.assertEqual(f, g.expand()) p = plot(*fs, ylim=(-10, 10), legend=False, show=False) colors = ['red', 'green', 'blue', 'brown', 'orange', 'purple', 'pink', 'gray', 'skyblue', 'yellow'] for i, s in enumerate(p): s.line_color = colors[i] for o in zip(p, colors): pprint(o) for i, f in enumerate(fs, 1): print(f'({i})') pprint(f.factor(gaussian=True)) p.show() p.save('sample28.png') if __name__ == "__main__": main()

% ./sample28.py -v 28. (cartesian line: 56*x**2 + 89*x + 35 for x over (-10.0, 10.0), red) (cartesian line: 3*x**2 + 2*x - 4 for x over (-10.0, 10.0), green) (cartesian line: 24*x**2 - 74*x - 35 for x over (-10.0, 10.0), blue) (cartesian line: 3*x**2 - 4*x - 2 for x over (-10.0, 10.0), brown) (cartesian line: 9*x**2 + 25 for x over (-10.0, 10.0), orange) (cartesian line: x**2 - x + 1 for x over (-10.0, 10.0), purple) (cartesian line: 2*x**2 - 14*sqrt(2)*x + 13 for x over (-10.0, 10.0), pink) (cartesian line: sqrt(5)*x**2 - sqrt(11)*x + sqrt(5) for x over (-10.0, 10.0), g ray) (1) 56⋅(x + 5/7)⋅(x + 7/8) (2) ⎛ 2 2⋅x 4⎞ 3⋅⎜x + ─── - ─⎟ ⎝ 3 3⎠ (3) 24⋅(x - 7/2)⋅(x + 5/12) (4) ⎛ 2 4⋅x 2⎞ 3⋅⎜x - ─── - ─⎟ ⎝ 3 3⎠ (5) ⎛ 5⋅ⅈ⎞ ⎛ 5⋅ⅈ⎞ 9⋅⎜x - ───⎟⋅⎜x + ───⎟ ⎝ 3 ⎠ ⎝ 3 ⎠ (6) 2 x - x + 1 (7) ⎛ 2 13⎞ 2⋅⎜x - 7⋅√2⋅x + ──⎟ ⎝ 2 ⎠ (8) 2 √5⋅x - √11⋅x + √5 test (__main__.MyTestCase) ... (1) (2) (3) (4) (5) (6) (7) (8) ok ---------------------------------------------------------------------- Ran 1 test in 0.012s OK %

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数学 - Python - 代数学 - 1次方程式, 2次方程式 - 解の係数の関係、2次式の因数分解 - 複素数の範囲

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