一道数学题,已知x+y+z=1,x^2+y^2+z^2=2,问xy+yz+zx,x^3+y^3+z^3
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一道数学题,已知x+y+z=1,x^2+y^2+z^2=2,问xy+yz+zx,x^3+y^3+z^3
xy+yz+xz={(x²+y²+z²+2xy+2xz+2yz)-(x²+y²+z²)}\2={(x+y+z)²-(x²+y²+z²)}\2=-1\2
(x+y+z)³=x³+y³+z³+2x²(y+z)+2y²(x+z)+2z²(x+y)
(x+y+z)(x²+y²+z²)= x³+y³+z³+x²(y+z)+y²(x+z)+z²(x+y)
x²(y+z)+y²(x+z)+z²(x+y)=(x+y+z)³-(x+y+z)(x²+y²+z²)=1-2=-1
x³+y³+z³=(x+y+z)(x²+y²+z²)- {x²(y+z)+y²(x+z)+z²(x+y)}=3
(x+y+z)³=x³+y³+z³+2x²(y+z)+2y²(x+z)+2z²(x+y)
(x+y+z)(x²+y²+z²)= x³+y³+z³+x²(y+z)+y²(x+z)+z²(x+y)
x²(y+z)+y²(x+z)+z²(x+y)=(x+y+z)³-(x+y+z)(x²+y²+z²)=1-2=-1
x³+y³+z³=(x+y+z)(x²+y²+z²)- {x²(y+z)+y²(x+z)+z²(x+y)}=3
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