已知x+y+z=1 x2+y2+z2=2 x3+y3+z3=3 求x4+y4+z4=?
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已知x+y+z=1 x2+y2+z2=2 x3+y3+z3=3 求x4+y4+z4=?
(x+y+z)²-(x²+y²+z²)=2(xy+yz+zx)=-1,xy+yz+zx=-1/2
x3+y3+z3=3xyz+(x+y+z)(x²+y²+z²-xy-yz-zx)=3xyz+1*(2-(-1/2))=3,xyz=1/6
(x2+y2+z2)²=x4+y4+z4-2x²y²-2y²z²-2x²z²=4,故x4+y4+z4=4+2(x²y²+y²z²+x²z²)
x²y²+y²z²+x²z²=(xy+yz+zx)²-2xyz(x+y+z)=(-1/2)²-2*1/6*1=-1/12
x4+y4+z4=4+2*(-1/12)=23/6
不清的话就追问.晚上好!
x3+y3+z3=3xyz+(x+y+z)(x²+y²+z²-xy-yz-zx)=3xyz+1*(2-(-1/2))=3,xyz=1/6
(x2+y2+z2)²=x4+y4+z4-2x²y²-2y²z²-2x²z²=4,故x4+y4+z4=4+2(x²y²+y²z²+x²z²)
x²y²+y²z²+x²z²=(xy+yz+zx)²-2xyz(x+y+z)=(-1/2)²-2*1/6*1=-1/12
x4+y4+z4=4+2*(-1/12)=23/6
不清的话就追问.晚上好!
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