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分解因式：(x^2-y^2)(1+zx)(1+zy)+(y^2-z^2)(1+xy)(1+xz)+(z^2-x^2)(1+yz)(1+yx)
题目详情
分解因式：(x^2-y^2)(1+zx)(1+zy)+(y^2-z^2)(1+xy)(1+xz)+(z^2-x^2)(1+yz)(1+yx)
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答案和解析
(x^2-y^2)(1+zx)(1+zy)+(y^2-z^2)(1+xy)(1+xz)+(z^2-x^2)(1+yz)(1+yx)
=(x^2-y^2)(1+zx+zy+xyz^2)+(y^2-z^2)(1+xy+xz+x^2yz)+(z^2-x^2)(1+yz+yx+xy^2z)
=(x^2-y^2)+(x^2-y^2)(zx+zy+xyz^2)+(y^2-z^2)+(y^2-z^2)(xy+xz+x^2yz)+(z^2-x^2)+(z^2-x^2)(yz+yx+xy^2z)
=z(x^2-y^2)(x+y+xyz)+x(y^2-z^2)(y+z+xyz)+y(z^2-x^2)(z+x+xyz)
=z(x^2-y^2)(x+y+z+xyz-z)+x(y^2-z^2)(x+y+z+xyz-x)+y(z^2-x^2)(x+y+z+xyz-y)
=z(x^2-y^2)(x+y+z+xyz)-z^2(x^2-y^2)+x(y^2-z^2)(x+y+z+xyz)-x^2(y^2-z^2)
+y(z^2-x^2)(x+y+z+xyz)-y^2(z^2-x^2)
=(z(x^2-y^2)+x(y^2-z^2)+y(z^2-x^2))(x+y+z+xyz)-z^2(x^2-y^2)-x^2(y^2-z^2)-y^2(z^2-x^2)
=(x^2z-y^2z+xy^2-xz^2+yz^2-x^2y)(x+y+z+xyz)-x^2z^2+y^2z^2-x^2y^2+x^2z^2-y^2z^2+x^2y^2
=(x^2z-y^2z+xy^2-xz^2+yz^2-x^2y)(x+y+z+xyz)
=(xyz-xyz+x^2z-y^2z+xy^2-xz^2+yz^2-x^2y)(x+y+z+xyz)
=(xyz-y^2z-xyz+yz^2+x^2z-xz^2+xy^2-x^2y)(x+y+z+xyz)
=(yz(x-y)-yz(x-z)+xz(x-z)-xy(x-y))(x+y+z+xyz)
=(yz(x-y)+(x-y)z(x-z)-xy(x-y))(x+y+z+xyz)
=(x-y)(yz+z(x-z)-xy)(x+y+z+xyz)
=(x-y)(z(x-z)-y(x-z))(x+y+z+xyz)
=(x-y)(x-z)(z-y)(x+y+z+xyz)
=(x^2-y^2)(1+zx+zy+xyz^2)+(y^2-z^2)(1+xy+xz+x^2yz)+(z^2-x^2)(1+yz+yx+xy^2z)
=(x^2-y^2)+(x^2-y^2)(zx+zy+xyz^2)+(y^2-z^2)+(y^2-z^2)(xy+xz+x^2yz)+(z^2-x^2)+(z^2-x^2)(yz+yx+xy^2z)
=z(x^2-y^2)(x+y+xyz)+x(y^2-z^2)(y+z+xyz)+y(z^2-x^2)(z+x+xyz)
=z(x^2-y^2)(x+y+z+xyz-z)+x(y^2-z^2)(x+y+z+xyz-x)+y(z^2-x^2)(x+y+z+xyz-y)
=z(x^2-y^2)(x+y+z+xyz)-z^2(x^2-y^2)+x(y^2-z^2)(x+y+z+xyz)-x^2(y^2-z^2)
+y(z^2-x^2)(x+y+z+xyz)-y^2(z^2-x^2)
=(z(x^2-y^2)+x(y^2-z^2)+y(z^2-x^2))(x+y+z+xyz)-z^2(x^2-y^2)-x^2(y^2-z^2)-y^2(z^2-x^2)
=(x^2z-y^2z+xy^2-xz^2+yz^2-x^2y)(x+y+z+xyz)-x^2z^2+y^2z^2-x^2y^2+x^2z^2-y^2z^2+x^2y^2
=(x^2z-y^2z+xy^2-xz^2+yz^2-x^2y)(x+y+z+xyz)
=(xyz-xyz+x^2z-y^2z+xy^2-xz^2+yz^2-x^2y)(x+y+z+xyz)
=(xyz-y^2z-xyz+yz^2+x^2z-xz^2+xy^2-x^2y)(x+y+z+xyz)
=(yz(x-y)-yz(x-z)+xz(x-z)-xy(x-y))(x+y+z+xyz)
=(yz(x-y)+(x-y)z(x-z)-xy(x-y))(x+y+z+xyz)
=(x-y)(yz+z(x-z)-xy)(x+y+z+xyz)
=(x-y)(z(x-z)-y(x-z))(x+y+z+xyz)
=(x-y)(x-z)(z-y)(x+y+z+xyz)
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