已知x+y+z=0,xyz不等于 0,求x(y分之1 +z分之1 )+y(x分之1 +z分之1 )+z(x分之1 +y分之1 )的值?
问题描述：
已知x+y+z=0,xyz不等于 0,求x(y分之1 +z分之1 )+y(x分之1 +z分之1 )+z(x分之1 +y分之1 )的值?
答
x+y+z=0 x+y=-z x+z=-y y+z=-x
(x+y+z)^3=0
x^3+y^3+z^3+3x^2y+3x^2z+3y^2x+3y^2z+3z^2x+3z^2y+6xyz=0
x^3+y^3+z^3=-(3x^2y+3x^2z+3y^2x+3y^2z+3z^2x+3z^2y+6xyz)
x(1/y+1/z)+y(1/x+1/z)+z(1/x+1/y)
=x(y+z)/yz+y(x+z)/xz+z(x+y)/xy
=-x^2/yz+(-y^2)/xz+(-z^2)/xy=-(x^3+y^3+z^3)/xyz
=3x^2y+3x^2z+3y^2x+3y^2z+3z^2x+3z^2y+6xyz/xyz
=3x/z+3x/y+3y/z+3y/x+3z/x+3z/y+6
=3(x+y)/z+3(y+z)/x+3(z+x)/y+6
=-3-3-3+6
=-3