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初一因式分解题x(y－z)^3＋y(z－x)^3+z(x－y)^3
题目详情
初一因式分解题
x(y－z)^3＋y(z－x)^3+z(x－y)^3
x(y－z)^3＋y(z－x)^3+z(x－y)^3
▼优质解答
答案和解析
x(y－z)^3＋y(z－x)^3+z(x－y)^3
=xy^3-3xy^2z+3xyz^2-xz^3+yz^3-3xyz^2+3x^2yz-x^3y+x^3z-3x^2yz+3xy^2z-y^3z
=xy^3-xz^3+yz^3-x^3y+x^3z-y^3z
=xy^3-y^3z+x^3z-xz^3-x^3y+yz^3
=y^3(x-z)+xz(x^2-z^2)-y(x^3-z^3)
=y^3(x-z)+xz(x+z)(x-z)-y(x-z)(x^2+xz+z^2)
=(x-z)(y^3+x^2z+xz^2-x^2y-xyz-yz^2)
=(x-z)(y^3-x^2y+x^2z-xyz+xz^2-yz^2)
=(x-z)(y-x)(y^2+xy-xz-z^2)
=(x-z)(y-x)[(y-z)(y+z)+x(y-z)]
=(x-z)(y-x)(y-z)(x+y+z)
=xy^3-3xy^2z+3xyz^2-xz^3+yz^3-3xyz^2+3x^2yz-x^3y+x^3z-3x^2yz+3xy^2z-y^3z
=xy^3-xz^3+yz^3-x^3y+x^3z-y^3z
=xy^3-y^3z+x^3z-xz^3-x^3y+yz^3
=y^3(x-z)+xz(x^2-z^2)-y(x^3-z^3)
=y^3(x-z)+xz(x+z)(x-z)-y(x-z)(x^2+xz+z^2)
=(x-z)(y^3+x^2z+xz^2-x^2y-xyz-yz^2)
=(x-z)(y^3-x^2y+x^2z-xyz+xz^2-yz^2)
=(x-z)(y-x)(y^2+xy-xz-z^2)
=(x-z)(y-x)[(y-z)(y+z)+x(y-z)]
=(x-z)(y-x)(y-z)(x+y+z)
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