已知X+Y+Z=0,求证X^3+Y^3+Z^3=3XYZ
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已知X+Y+Z=0,求证X^3+Y^3+Z^3=3XYZ已知X+Y+Z=0,求证X^3+Y^3+Z^3=3XYZ已知X+Y+Z=0,求证X^3+Y^3+Z^3=3XYZ题意即要求证X^3+Y^3+Z^
已知X+Y+Z=0,求证X^3+Y^3+Z^3=3XYZ
已知X+Y+Z=0,求证X^3+Y^3+Z^3=3XYZ
已知X+Y+Z=0,求证X^3+Y^3+Z^3=3XYZ
题意即要求证X^3+Y^3+Z^3-3XYZ=0
证明如下:x^3+y^3+z^3-3xyz =(x+y+z)(x^2+y^2+z^2-xy-yz-zx)
因为X+Y+Z=0,所以上式等于0.所以x^3+y^3+z^3-3xyz=0,X^3+Y^3+Z^3=3XYZ,得证.
x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)
由x^3+y^3+z^3=(x+y+z)(x^2+y^2+z^2)-z(x^2+y^2)-x(y^2+z^2)-y(x^2+z^2)
x^3+y^3+z^3-3xyz=……=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)
因为X+Y+Z=0
所以:x^3+y^3+z^3-3xyz=0
所以:X^3+Y^3+Z^3=3XYZ
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