第1个回答 2009-12-04
(x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2xz.....
(x+y+z)(x^2+y^2+z^2)=x^3+y^3+z^3+xy(x+y)+xz(x+z)+yz(y+z)
=3+xy(1-z)+xz(1-y)+yz(1-x)
=3+xy+yz+xz-3xyz
xy+yz+xz=-1/2
原式=3+(-1/2)-3xyz=2 xyz=1/6
(x+y+z)(x^3+y^3+z^3)=x^4+y^4+z^4+xy(x^2+y^2)+yz(y^2+z^2)+xz(x^2+z^2)
=x^4+y^4+z^4+xy(1-z^2)+yz(1-x^2)+xz(1-y^2)
=x^4+y^4+z^4+xy+yz+xz-xyz(x+y+z)
=x^4+y^4+z^4+(-1/2)-1/6
=3
x^4+y^4+z^4=3+1/2+1/6