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x^3+y^3+z^3-3xyz
题目详情
x^3+y^3+z^3-3xyz
▼优质解答
答案和解析
x^3+y^3+z^3-3xyz
= (x^3+3yx^2+3xy^2+y^3)+z^3-3xyz-3yx^2-3xy^2
= (x+y)^3+z^3-3xy(x+y+z)
= (x+y+z)[(x+y)^2-(x+y)z+z^2]-3xy(x+y+z)
= (x+y+z)[(x^2+2xy+y^2-xz-yz+z^2)-2xy]
= (x+y+z)(x^2+y^2+z^2-xy-yz-zx)
= (x^3+3yx^2+3xy^2+y^3)+z^3-3xyz-3yx^2-3xy^2
= (x+y)^3+z^3-3xy(x+y+z)
= (x+y+z)[(x+y)^2-(x+y)z+z^2]-3xy(x+y+z)
= (x+y+z)[(x^2+2xy+y^2-xz-yz+z^2)-2xy]
= (x+y+z)(x^2+y^2+z^2-xy-yz-zx)
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