cho x,y,z thuộc Z chứng minh (x-y)^5+(y-z)^5+(x-x)^5 chia hết cho 5(x-y)(y-z)(z-x)

Đáp án:Giải thích các bước giải:

Đặt(x−y;y−z;z−x)=(a;b;c)⇒a+b+c=x−y+y−z+z−x=0

Khi đó : (x−y)5+(y−z)5+(z−x)5=a5+b5+c5 với a+b+c=0Do a+b+c=0⇔c=−(a+b)

⇔c5=-(a+b)5⇒a5+b5+c5=a5+b5(a+b)5=a5+b5−a5−5a4b−10a3b2−10a2b3−5ab4−b5

=−5a4b−10a3b2−10a2b3−5ab4

=−[5ab(a3+b3)+10a2b2(a+b)]

=−5[ab(a+b)(a2−ab+b2)+2a2b2(a+b)]

=−5ab(a+b)(a2−ab+b2+2ab)

=−5ab.(−c)(a2+ab+b2)

=5abc[a(−b−c)+ab+b(−a−c)]

= 5 a b c ( − a b − a c + a b − a b − b c ) ]

= − 5 a b c ( a b + a c + b c ) 5 a b c Hay   ( x − y ) 5 + ( y − z ) 5 + ( z − x ) 5 5 ( x − y ) ( y − z ) ( z − x )   ( dpcm )

 Vậy  ( x − y ) 5 + ( y − z ) 5 + ( z − x ) 5 5 ( x − y ) ( y − z ) ( z − x )