已知X+Y+Z=0,求证X＾3 +Y＾3+Z＾3＝3XYZ
问题描述：
已知X+Y+Z=0,求证X＾3 +Y＾3+Z＾3＝3XYZ
答
X+Y+Z=0
(X+Y+Z)^3 = 0
(X²+Y²+Z²+2XY+2XZ+2YZ )(X+Y+Z) = 0
(X²+Y²+Z²) (X+Y+Z) = 0
即：X³+Y³+Z³+X²Y+X²Z+XY²+Y²Z+XZ²+YZ² = 0…………………………1式
将(X+Y+Z)^3=0完全展开得：
X³+Y³+Z³+3X²Y+3X²Z+3XY²+3Y²Z+3XZ²+3YZ²+6XYZ = 0……………2式
1式乘3,再减去2式,得：
2X³+2Y³+2Z³ = 6xyz
即：X³+Y³+Z³ = 3xyz,得证