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已知x+y+z=0,x^2+y^2+z^2=1,试求xy+yz+zx和x^4+y^4+z^4的值
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已知x+y+z=0,x^2+y^2+z^2=1,试求xy+yz+zx和x^4+y^4+z^4的值
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答案和解析
1.(x+y+z)²-x²-y²-z²=2(xy+yx+zx)=-1
xy+yz+zx=-1/2
2.(xy+yz+zx)²=x²y²+y²z²+z²x²+2xyz(x+y+z)
=x²y²+y²z²+z²x²+0=1/4
(x²+y²+z²)²=x^4+y^4+z^4+2×(x²y²+y²z²+z²x²)
x^4+y^4+z^4=1-1/2=1/2
xy+yz+zx=-1/2
2.(xy+yz+zx)²=x²y²+y²z²+z²x²+2xyz(x+y+z)
=x²y²+y²z²+z²x²+0=1/4
(x²+y²+z²)²=x^4+y^4+z^4+2×(x²y²+y²z²+z²x²)
x^4+y^4+z^4=1-1/2=1/2
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