数学 – Python - 数学はここから始まる - 数 – 整式 – 共通因数をくくり出すこと(因数分解、応用、色々な工夫)

数学読本〈1〉

学習環境

• Surface Go、タイプ カバー、ペン(端末)
• Windows 10 Pro (OS)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo(iPad アプリ(iOS))
• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

数学読本〈1〉数・式の計算/方程式/不等式 (松坂 和夫(著)、岩波書店)の第2章(文字と記号の活躍 - 式の計算)、2.1(整式)、共通因数をくくり出すことの問13-(1)、(2)、(3)、(4)、(5)、(6)、(7)、(8)、(9)、(10)、(11)、(12)、(13)、(14)、(15)、(16)、(17)、(18)、(19)、(20).を取り組んでみる。

13. 
    
    1. 
       
       $\begin{array}{}\left(x+y\right)\left(x-y\right)-\left(x+y\right)\\ =\left(x+y\right)\left(x-y-1\right)\end{array}$
    2. 
       
       $\begin{array}{}\left(a+c\right)\left(a-c\right)+b\left(a-c\right)\\ =\left(a+b+c\right)\left(a-c\right)\end{array}$
    3. 
       
       $\begin{array}{}{\left(x-y\right)}^2+5\left(x-y\right)+6\\ =\left(x-y+2\right)\left(x-y+3\right)\end{array}$
    4. 
       
       $\begin{array}{}{x}^2\left(x-y\right)-\left(x-y\right)\\ =\left(x^2-1\right)\left(x-y\right)\\ =\left(x+1\right)\left(x-1\right)\left(x-y\right)\end{array}$
    5. 
       
       $\begin{array}{}{\left(x^2-1\right)}^2\\ ={\left(x+1\right)}^2{\left(x-1\right)}^2\end{array}$
    6. 
       
       $\begin{array}{}\left(x^2-y^2\right)\left(x^2-25y^2\right)\\ =\left(x+y\right)\left(x-y\right)\left(x+5y\right)\left(x-5y\right)\end{array}$
    7. 
       
       $\begin{array}{}\left(a^2+5\right)\left(a^2-4\right)\\ =\left(a^2+5\right)\left(a+2\right)\left(a-2\right)\end{array}$
    8. 
       
       $\begin{array}{}\left(a^2-4b^2\right)\left(a^2+4b^2\right)\\ =\left(a+2b\right)\left(a-2b\right)\left(a^2+4b^2\right)\end{array}$
    9. 
       
       $\begin{array}{}\left(4x^2+9y^2\right)\left(4x^2-9y^2\right)\\ =\left(4x^2+9y^2\right)\left(2x+3y\right)\left(2x-3y\right)\end{array}$
    10. 
        
        $\begin{array}{}\left(x^2+2y^2\right)\left(x^2-y^2\right)\\ =\left(x^2+2y^2\right)\left(x+y\right)\left(x-y\right)\end{array}$
    11. 
        
        $\begin{array}{}\left(x^2+4x-12\right)\left(x^2+4x+4\right)\\ =\left(x+6\right)\left(x-2\right){\left(x+2\right)}^2\end{array}$
    12. 
        
        $\begin{array}{}2x^2+\left(3y-4\right)x-\left(2y^2-7y+6\right)\\ =2x^2+\left(3y-4\right)x-\left(y-2\right)\left(2y-3\right)\\ =\left(x+2y-3\right)\left(2x-y+2\right)\end{array}$
    13. 
        
        $\begin{array}{}{x}^2-\left(y+1\right)x-\left(6y^2-23y+20\right)\\ =x^2-\left(y+1\right)x-\left(2y-5\right)\left(3y-4\right)\\ =\left(x+2y-5\right)\left(x-3y+4\right)\end{array}$
    14. 
        
        $\begin{array}{}\left(x-2\right)y+\left(2x^2-x-6\right)\\ =\left(x-2\right)y+\left(x-2\right)\left(2x+3\right)\\ =\left(x-2\right)\left(2x+y+3\right)\end{array}$
    15. 
        
        $\begin{array}{}{a}^2+\left(2b-3\right)a-\left(b+1\right)\left(3b-2\right)\\ =\left(a-b-1\right)\left(a+3b-2\right)\end{array}$
    16. 
        
        $\begin{array}{}{\left(a^2+2\right)}^2-4a^2\\ =\left(a^2+2a+2\right)\left(a^2-2a+2\right)\end{array}$
    17. 
        
        $\begin{array}{}{\left(x^2+1\right)}^2-x^2\\ =\left(x^2+x+1\right)\left(x^2-x+1\right)\end{array}$
    18. 
        
        $\begin{array}{}{\left(a+1\right)}^2+\left(b+c\right)\left(a+1\right)+bc\\ =\left(a+b+1\right)\left(a+c+1\right)\end{array}$
    19. 
        
        $\begin{array}{}{\left(x+y\right)}^3-3x^{2}y-3x{y}^2+1-3xy\\ =\left(x+y+1\right)\left({\left(x+y\right)}^2-\left(x+y\right)+1\right)-3xy\left(x+y+1\right)\\ =\left(x+y+1\right)\left(x^2+y^2+1-y-x-xy\right)\end{array}$
    20. 
        
        $\begin{array}{}{\left(a-b\right)}^3+{\left(b-c\right)}^3+{\left(c-a\right)}^3-3\left(a-b\right)\left(b-c\right)\left(c-a\right)+3\left(a-b\right)\left(b-c\right)\left(c-a\right)\\ =0+3\left(a-b\right)\left(b-c\right)\left(c-a\right)\\ =3\left(a-b\right)\left(b-c\right)\left(c-a\right)\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols

print('13.')

x, y, a, b, c = symbols('x, y, a, b, c', real=True)

ts = [x ** 2 - x - y ** 2 - x,
      a ** 2 - c ** 2 + a * b - b * c,
      (x - y) * (x - y + 5) + 6,
      x ** 3 - x ** 2 * y - x + y,
      x ** 4 - 2 * x ** 2 + 1,
      x ** 4 - 26 * x ** 2 * y ** 2 + 25 * y ** 4,
      a ** 4 + a ** 2 - 20,
      a ** 4 - 16 * b ** 4,
      16 * x ** 4 - 81 * y ** 4,
      x ** 4 + x ** 2 * y ** 2 - 2 * y ** 4,
      (x ** 2 + 4 * x) ** 2 - 8 * (x ** 2 + 4 * x) - 48,
      2 * x ** 2 + 3 * x * y - 2 * y ** 2 - 4 * x + 7 * y - 6,
      x ** 2 - x * y - 6 * y ** 2 - x + 23 * y - 20,
      2 * x ** 2 + x * y - x - 2 * y - 6,
      a ** 2 + (2 * b - 3) * a - (3 * b ** 2 + b - 2),
      a ** 4 + 4,
      x ** 4 + x ** 2 + 1,
      (a + b + c + 1) * (a + 1) + b * c,
      x ** 3 + y ** 3 + 1 - 3 * x * y,
      (a - b) ** 3 + (b - c) ** 3 + (c - a) ** 3]

for i, t in enumerate(ts, 1):
    print(f'({i})')
    pprint(t.factor())
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample13.py
13.
(1)
 2          2
x  - 2⋅x - y 

(2)
(a - c)⋅(a + b + c)

(3)
(x - y + 2)⋅(x - y + 3)

(4)
(x - 1)⋅(x + 1)⋅(x - y)

(5)
       2        2
(x - 1) ⋅(x + 1) 

(6)
(x - 5⋅y)⋅(x - y)⋅(x + y)⋅(x + 5⋅y)

(7)
                ⎛ 2    ⎞
(a - 2)⋅(a + 2)⋅⎝a  + 5⎠

(8)
                    ⎛ 2      2⎞
(a - 2⋅b)⋅(a + 2⋅b)⋅⎝a  + 4⋅b ⎠

(9)
                        ⎛   2      2⎞
(2⋅x - 3⋅y)⋅(2⋅x + 3⋅y)⋅⎝4⋅x  + 9⋅y ⎠

(10)
                ⎛ 2      2⎞
(x - y)⋅(x + y)⋅⎝x  + 2⋅y ⎠

(11)
               2        
(x - 2)⋅(x + 2) ⋅(x + 6)

(12)
(x + 2⋅y - 3)⋅(2⋅x - y + 2)

(13)
(x - 3⋅y + 4)⋅(x + 2⋅y - 5)

(14)
(x - 2)⋅(2⋅x + y + 3)

(15)
(a - b - 1)⋅(a + 3⋅b - 2)

(16)
⎛ 2          ⎞ ⎛ 2          ⎞
⎝a  - 2⋅a + 2⎠⋅⎝a  + 2⋅a + 2⎠

(17)
⎛ 2        ⎞ ⎛ 2        ⎞
⎝x  - x + 1⎠⋅⎝x  + x + 1⎠

(18)
(a + b + 1)⋅(a + c + 1)

(19)
            ⎛ 2              2        ⎞
(x + y + 1)⋅⎝x  - x⋅y - x + y  - y + 1⎠

(20)
-3⋅(a - b)⋅(a - c)⋅(b - c)

$