Write $x^2y+xy^2+x^2z+xz^2+y^2z+yz^2 $ as a product of elementary symmetric polynomial
I get $E1=x+y+z$, $E2=xy+xz+yz$, $E3=xyz$. I've tried factoring out E3(xyz) but I can tell that's not right. I know this probably isn't that difficult, think I'm just going about it the wrong way.
Please help!!
There's a unique way to write $x^2y+xy^2+x^2z+xz^2+y^2z+yz^2$ as a polynomial in $E_1,E_2,E_3$. Note that $x^2y$ could nicely be obtained by expanding the product $(x+\ldots)(xy+\ldots)$, and indeed after expanding $E_1E_2$, we obtain the desired expression - plus $3xyz$. Hence $$x^2y+xy^2+x^2z+xz^2+y^2z+yz^2=E_1E_2-3E_3.$$