I was going through a proof and couldn't understand this part about homogeneous symmetric polynomials.
$$x^4(y − z) + y^4(z − x) + z^4(x − y) = −(x − y)(y − z)(z − x) $$
Viewing the left-hand side as a polynomial in x, the zeros of the polynomial are y and z, and its coefficients are divisible by y − z. Hence there is a quadratic homogeneous symmetric polynomial Q(x, y, z) such that
$$ x^4 (y − z) + y^4 (z − x) + z^4 (x − y) = (x − y)(y − z)(z − x)Q(x, y,z) $$
I know what a homogeneous symmetric polynomial is, but I was suspecting that there was some kind of underlying theorem used that wasn't outlined explicitly.
