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

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
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

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