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2018年9月27日木曜日

学習環境

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。



    1. (x+y)(x-y)-(x+y)=(x+y)(x-y-1)

    2. (a+c)(a-c)+b(a-c)=(a+b+c)(a-c)

    3. (x-y)2+5(x-y)+6=(x-y+2)(x-y+3)

    4. x2(x-y)-(x-y)=(x2-1)(x-y)=(x+1)(x-1)(x-y)

    5. (x2-1)2=(x+1)2(x-1)2

    6. (x2-y2)(x2-25y2)=(x+y)(x-y)(x+5y)(x-5y)

    7. (a2+5)(a2-4)=(a2+5)(a+2)(a-2)

    8. (a2-4b2)(a2+4b2)=(a+2b)(a-2b)(a2+4b2)

    9. (4x2+9y2)(4x2-9y2)=(4x2+9y2)(2x+3y)(2x-3y)

    10. (x2+2y2)(x2-y2)=(x2+2y2)(x+y)(x-y)

    11. (x2+4x-12)(x2+4x+4)=(x+6)(x-2)(x+2)2

    12. 2x2+(3y-4)x-(2y2-7y+6)=2x2+(3y-4)x-(y-2)(2y-3)=(x+2y-3)(2x-y+2)

    13. x2-(y+1)x-(6y2-23y+20)=x2-(y+1)x-(2y-5)(3y-4)=(x+2y-5)(x-3y+4)

    14. (x-2)y+(2x2-x-6)=(x-2)y+(x-2)(2x+3)=(x-2)(2x+y+3)

    15. a2+(2b-3)a-(b+1)(3b-2)=(a-b-1)(a+3b-2)

    16. (a2+2)2-4a2=(a2+2a+2)(a2-2a+2)

    17. (x2+1)2-x2=(x2+x+1)(x2-x+1)

    18. (a+1)2+(b+c)(a+1)+bc=(a+b+1)(a+c+1)

    19. (x+y)3-3x2y-3xy2+1-3xy=(x+y+1)((x+y)2-(x+y)+1)-3xy(x+y+1)=(x+y+1)(x2+y2+1-y-x-xy)

    20. (a-b)3+(b-c)3+(c-a)3-3(a-b)(b-c)(c-a)+3(a-b)(b-c)(c-a)=0+3(a-b)(b-c)(c-a)=3(a-b)(b-c)(c-a)

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)

$

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