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 7 ab  43 abab 2()()()3 ()()  0  a  b ab , xaxb  0 t 

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[ 東京工業大学 1965 年 ２ ]

三次方程式

x

3

ax b   0

の

3

根を

   , ,

とし，

t

n

 

n

 

n

 

n とおくとき，

at

5

bt

4を

,

a b

で表せ。

解と係数の関係より

      0

，

      a

，

   b

よって

2

 

2

 

2

 (      )

2

 2(      )   2a

ここで，

ax

5

bx

4

 ( x

3

ax b ax  )(

2

bx )  a x

2 3

 2 abx

2

b x

2

3 2 2 3 3 2 2 2

( x ax b ax )( bx ) a x ( ax b ) a x a b 2 abx b x

          

3 2 2 3 2 3 2 2

( x ax b ax )( bx ) a x ( ax b ) 2 abx ( a b x ) a b

          

となることから

5 4 2 3 2 2

2 ( )

a   b    ab   ab   a b

5 4 2 3 2 2

2 ( )

a   b    ab   ab   a b

5 4 2 3 2 2

2 ( )

a   b    ab   ab   a b

よって

at

5

bt

4

a ( 

5

 

5

 

5

)  b ( 

4

 

4

 

4

)

2 2 2 3 2 2

2 ab (    ) ( a b )(    ) 3 a b

        

2 2

4 a b 3 a b

 

7a b

2

参照

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