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)

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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)
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)

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


x(1/y+1/z)+y(1/x+1/z)+z(1/x+1/y)
=x²(y+z)/xyz+y²(x+z)/xyz+z²(x+y)/xyz
=(x²y+x²z+y²x+y²z+z²x+z²y)/xyz
=[xy(x+y)+xz(x+z)+yz(y+z)]/xyz
∵x...

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x(1/y+1/z)+y(1/x+1/z)+z(1/x+1/y)
=x²(y+z)/xyz+y²(x+z)/xyz+z²(x+y)/xyz
=(x²y+x²z+y²x+y²z+z²x+z²y)/xyz
=[xy(x+y)+xz(x+z)+yz(y+z)]/xyz
∵x+y+z=0
∴x+y=-z
y+z=-x
x+z=-y

x(1/y+1/z)+y(1/x+1/z)+z(1/x+1/y)
=[xy(x+y)+xz(x+z)+yz(y+z)]/xyz
=(-xyz-xyz-xyz)/xyz
=-3

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