Q:

Prove that $f(x)y+g(x)$ does not factor as a product of two polynomials of positive degree.

Let $f(x)$ and $g(x)$ be polynomials  that have no common factor of positive degree. Prove that $f(x)y+g(x)$ does not factor as a product of two polynomials of positive degree. 
I am now learning intersection multiplicity at the origin of two polynomials. I don't know how to prove this using this part of knowledge. 
Thanks.

A:

Assuming we have polynomials over a field: if $f(x) y + g(x)$ were to factor into two polynomials, there must not be any $y^2$ terms, so it must factor as $(a(x) y + b(x)) c(x)$. Then it follows that $a(x) c(x) = f(x)$ and $b(x) c(x) = g(x)$. So $c(x)$ has degree $0$.
Thus if $f(x) y + g(x)$ factors into two polynomials, one of them has degree $0$.

