因式分解: x^3+y^3+z^3- 3xyz

因式分解: x^3+y^3+z^3-3xyz

原連結


 

 

X^3 + Y^3 + Z^3- 3XYZ
=(x+y)^3-3x^y-3xy^2+z^3-3xyz
=(x+y)^3+z^3-3x^y-3xy^2-3xyz
=(x+y+z)^3-3(x+y)^2z-3(x+y)z^2-3xy(x+y)-3xyz
=(x+y+z)^3-3[(x+y)^2z+(x+y)z^2+xy(x+y)+xyz]
=(x+y+z)^3-3[(x+y)z(x+y+z)+xy(x+y+z)]
=(x+y+z)^3-3(x+y+z)[(x+y)z+xy]
=(x+y+z)[(x+y+z)^2-3(xz+yz+xy)]
=(x+y+z)(x^2+y^2+z^2-xz-yz-xy)

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