试证明(x+y-2z)³+(y+z-2x)³+(z+x-2y)³=3(x+y-2z)(y+z-2x)(z+x-2y)

试证明(x+y-2z)³+(y+z-2x)³+(z+x-2y)³=3(x+y-2z)(y+z-2x)(z+x-2y)
试证明(x+y-2z)³+(y+z-2x)³+(z+x-2y)³=3(x+y-2z)(y+z-2x)(z+x-2y)

试证明(x+y-2z)³+(y+z-2x)³+(z+x-2y)³=3(x+y-2z)(y+z-2x)(z+x-2y)
若 a+b+c=0,可证
a^3+b^3+c^3 - 3 abc
= a^3+b^3+(-a-b)^3 - 3 ab(-a-b)
= a^3+b^3 + 3 ab(a+b) - (a+b)^3
= (a+b)^3 - (a+b)^3
令 a= x+y-2z,b= y+z-2x,c= z+x-2y ,即得.