Question

因式分解(x+y+z)^3-(y+z-x)^3-(z+x-y)^3-(x+y-z)^3

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Answer 1

x3-y3=(x-y)(x2+xy+y2) ①
x3+y3=(x+y)(x2-xy+y2) ②
(x+y+z)3-(y+z-x)3=2x[(x+y+z)2+(y+z)2-x2+(y+z-x)2] ③
(z+x-y)3+(x+y-z)3=2x[(z+x-y)2-x2+(y-z)2+(x+y-z)2] ④
3和4相减,相减之后用平方差公式～（1和2是立方和差公式）
2x[4y(x+z)+4yz+4y(z-x)]=2x*12yz=24xyz~

Answer 2

(x+y+z)^3-(y+z-x)^3-(z+x-y)^3-(x+y-z)^3
=((x+y+z)^3-(y+z-x)^3)-((z+x-y)^3+(x+y-z)^3)
=(x+y+z-y-z+x)((x+y+z)^2+(x+y+z)(y+z-x)+(y+z-x)^2)-
(z+x-y+x-z+y)((z+x-y)^2-(z+x-y)(x+y-z)+(x+y-z)^2)
=2x((x+y+z)^2+(x+y+z)(y+z-x)+(y+z-x)^2-(z+x-y)^2+(z+x-y)(x+y-z)-(x+y-z)^2)
=2x((x+y+z+z+x-y)(x+y+z-z-x+y)+(y+z-x+x+y-z)(y+z-x-x-y+z)+(x+y+z)(y+z-x)+(z+x-y)(x+y-z))
=2x(12yz)
=24xyz