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：
(y+z-2x)³+(z+x-2y)³+(x+y-2z)³
=[(y+z-2x)³+(z+x-2y)³]+(x+y-2z)³ /前面两项运用立方和公式
=[(y+z-2x)+(z+x-2y)][(y+z-2x)²-(y+z-2x)(z+x-2y)+(z+x-2y)²]+(x+y-2z)³
=(2z-x-y)[(y+z-2x)²+2(y+z-2x)(z+x-2y)+(z+x-2y)²-3(y+z-2x)(z+x-2y)]+(x+y-2z)³ /配方
=(x+y-2z)[3(y+z-2x)(z+x-2y)-[(y+z-2x)+(z+x-2y)]² ]+(x+y-2z)³
=(x+y-2z)[3(y+z-2x)(z+x-2y)-(2z-x-y)²]+(x+y-2z)³
=3(y+z-2x)(z+x-2y)(x+y-2z)-(x+y-2z)³+(x+y-2z)³
=3(y+z-2x)(z+x-2y)(x+y-2z)