Prove that `det ((yx-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.

Prove that `det ((yx-x^2,zx-y^2,xy-z^2),(zx-y^2,xy-z^2,yz-x^2),(xy-z^2

1.	Prove that `det ((yx-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.
Answer» `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]\|` `= (xy+yz+zx-x^2-y^2-z^2)(x+y+z)^2[(x-y)(y-x) - (x-z)(y-z)]` `=(xy+yz+zx-x^2-y^2-z^2)(x+y+z)^2[xy+yz+zx-x^2-y^2-z^2]` `Delta = (xy+yz+zx-x^2-y^2-z^2)^2(x+y+z)^2` `:. Delta` is divisible by `(x+y+z)` and quotient is `(xy+yz+zx-x^2-y^2-z^2)^2(x+y+z).`