Prove that `|[yz-x^2,zx-y^2,xy-z^2],[zx-y^2,xy-z^2,yz-x^2],[xy-z^2,yz-x^2,zx-y^2]|` is divisible by `(x+y+z),` and hence find the quotient.

Explanation

`Delta = |[yz-x^2,zx-y^2,xy-z^2],[zx-y^2,xy-z^2,yz-x^2],[xy-z^2,yz-x^2,zx-y^2]|`

Applying `C_1->C_1+C_2+C_3`

`Delta = |[xy+yz+zx-x^2-y^2-z^2,zx-y^2,xy-z^2],[xy+yz+zx-x^2-y^2-z^2,xy-z^2,yz-x^2],[xy+yz+zx-x^2-y^2-z^2,yz-x^2,zx-y^2]|`

`= (xy+yz+zx-x^2-y^2-z^2)|[1,zx-y^2,xy-z^2],[1,xy-z^2,yz-x^2],[1,yz-x^2,zx-y^2]|`

Now, applying `R_1->R_1-R_3` and `R_2->R_2-R_3`

`= (xy+yz+zx-x^2-y^2-z^2)|[0,zx-y^2-yz+x^2,xy-z^2-zx+y^2],[0,xy-z^2-yz+x^2,yz-x^2-zx+y^2],[1,yz-x^2,zx-y^2]|`

`= (xy+yz+zx-x^2-y^2-z^2)|[0,(x-y)(x+y+z),(y-z)(x+y+z)],[0,(x-z)(x+y+z),(y-x)(x+y+z)],[1,yz-x^2,zx-y^2]|`

`= (xy+yz+zx-x^2-y^2-z^2)(x+y+z)^2|[0,(x-y),(y-z)],[0,(x-z),(y-x)],[1,yz-x^2,zx-y^2]|`

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