证明:(y+z-2x)3+(z+x-2y)3+(x+y-2z)3=3(y+z-2x)(z+x-2y)(x+y-2z).

问题描述:

证明:(y+z-2x)3+(z+x-2y)3+(x+y-2z)3=3(y+z-2x)(z+x-2y)(x+y-2z).

证明:令y+z-2x=a,①
z+x-2y=b,②
x+y-2z=c,③
则要证的等式变为
a3+b3+c3=3abc.
联想到乘法公式:
a3+b3+c3-3abc=(a+b+c)(a2+b2+c2-ab-bc-ca),
∴将①,②,③相加有:a+b+c=y+z-2x+z+x-2y+x+y-2z=0,
∴a3+b3+c3-3abc=0,
∴(y+z-2x)3+(z+x-2y)3+(x+y-2z)3=3(y+z-2x)(z+x-2y)(x+y-2z).