2020年8月5日水曜日

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7章(不等式)の練習問題14.の解答を求めてみる。



    1. a 3 + b 3 - a 2 b - a b 2
      = ( a + b ) 3 - 3 a 2 b - 3 a b 2 - a 2 b - a b 2
      = ( a + b ) 3 - 4 a b ( a + b )
      = ( a + b ) ( ( a + b ) 2 - 4 a b )
      = ( a + b ) ( a 2 + b 2 - 2 a b )
      = ( a + b ) ( a - b ) 2
      0

      よって

      a 3 + b 3 a 2 b + a b 2

      (証明終)


    2. 2 ( a 5 + b 5 ) - ( a 2 + b 2 ) ( a 3 + b 3 )
      = 2 ( a 5 + b 5 ) - ( a 5 + a 2 b 3 + a 3 b 2 + b 5 )
      = a 5 + b 5 - a 2 b 3 - a 3 b 2
      = ( a + b ) 5 - 5 a 4 b - 10 a 3 b 2 - 10 a 2 b 3 - 5 a b 4 - a 2 b 3 - a 3 b 2
      = ( a + b ) 5 - 5 a 4 b - 11 a 3 b 2 - 11 a 2 b 3 - 5 a b 4
      = ( a + b ) 5 - a b ( 5 a 3 + 11 a 2 b + 11 a b 2 + 5 b 3 )
      = ( a + b ) 5 - a b ( 5 a 3 + 15 a 2 b + 15 a b 2 + 5 b 2 - 4 a 2 b - 4 a b 2 )
      = ( a + b ) 5 - a b ( 5 ( a + b ) 3 - 4 a b ( a + b ) )
      = ( a + b ) ( ( a + b ) 4 - a b ( 5 ( a + b ) 2 - 4 a b ) )
      = ( a + b ) ( a 4 + 4 a 3 b + 6 a 2 b 2 + 4 a b 3 + b 4 - a b ( 5 a 2 + 6 a b + 5 b 2 ) )
      = ( a + b ) ( a 4 - a 3 b - a b 3 + b 4 )
      = ( a + b ) ( a 3 ( a - b ) - b 3 ( a - b ) )
      = ( a + b ) ( a - b ) ( a 3 - b 3 )
      = ( a + b ) ( a - b ) 2 ( a 2 + a b + b 2 )
      0

      よって、

      2 ( a 5 + b 5 ) ( a 2 + b 2 ) ( a 3 + b 3 )

      (証明終)

コード

#!/usr/bin/env python3
from sympy.plotting import plot3d
from sympy.abc import a, b

print('14.')

p = plot3d(
    a ** 3 + b ** 3,
    a ** 2 + b + a * b ** 2,
    (a, 0, 5),
    (b, 0, 5),
    show=False
)
p.xlabel = a
p.ylabel = b
p.save('sample14_1.png')

p = plot3d(
    2 * (a ** 5 + b ** 5),
    (a ** 2 + b ** 2) * (a ** 3 + b ** 3),
    (a, 0, 5),
    (b, 0, 5),
    show=False
)
p.xlabel = a
p.ylabel = b
p.save('sample14_2.png')

p.show()

入出力結果(Zsh、PowerShell、Terminal、Jupyter(IPython))

% ./sample14.py
14.
%

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