已知x+y+z=3,x^2+y^2+z^2=29,x^3+y^3+z^3=45,求xyz的值.
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已知x+y+z=3,x^2+y^2+z^2=29,x^3+y^3+z^3=45,求xyz的值.
![已知x+y+z=3,x^2+y^2+z^2=29,x^3+y^3+z^3=45,求xyz的值.](/uploads/image/z/2772917-53-7.jpg?t=%E5%B7%B2%E7%9F%A5x%2By%2Bz%3D3%2Cx%5E2%2By%5E2%2Bz%5E2%3D29%2Cx%5E3%2By%5E3%2Bz%5E3%3D45%2C%E6%B1%82xyz%E7%9A%84%E5%80%BC.)
需要用到因式分解x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-xz)
分解过程参见:
(x+y+z)^2-(x^2+y^2+z^2)
=2xy+2xz+2yz
所以xy+xz+yz=[3*3-29]/2=-10
所以
3xyz=x^3+y^3+z^3-(x+y+z)(x^2+y^2+z^2-xy-yz-xz)
=45-3*[29-(-10)]
=-72
xyz=-24
分解过程参见:
(x+y+z)^2-(x^2+y^2+z^2)
=2xy+2xz+2yz
所以xy+xz+yz=[3*3-29]/2=-10
所以
3xyz=x^3+y^3+z^3-(x+y+z)(x^2+y^2+z^2-xy-yz-xz)
=45-3*[29-(-10)]
=-72
xyz=-24
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