若x+y+z=0,且xyz≠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)

问题描述:

若x+y+z=0,且xyz≠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)

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)
=x/y+x/z+y/x+y/z+z/x+z/y
=(x+z)/y + (x+y)/z + (y+z)/x
x+y+z=0
则:x+y=-z,y+z=-x ,x+z=-y
∴原式=(-1)+(-1)+(-1)
=-3