已知x+y+z=3,x^2+y^2+z^2=19,x^3+y^3+z^3=30 则xyz=

已知x+y+z=3,x^2+y^2+z^2=19,x^3+y^3+z^3=30 则xyz=
已知x+y+z=3,x^2+y^2+z^2=19,x^3+y^3+z^3=30 则xyz=

已知x+y+z=3,x^2+y^2+z^2=19,x^3+y^3+z^3=30 则xyz=
牢记一个基本的公式:
x^3+y^3+z^3-3xyz
=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)
=(x+y+z)[(x+y+z)^2-3(xy+yz+zx)]
所以:
根据x+y+z=3,两边平方,有:
x^2+y^2+z^2+2xy+2yz+2zx=9.
再有:x^2+y^2+z^2=19,所以:xy+yz+yz=-5.
所以代入上面的公式,有:
30-3xyz=3*[3*3-3*(-5)]
所以xyz=-14