Fulton's book on Algebraic Curve asks a question that, over a domain R, a factor of a form (by which he means a homogeneous polynomial) in indeterminates $x_1,.., x_n$ is also a form.
For one variable my approach one like this, Since $f=ax^n$ can be the only homogeneous polynomial of degree $n$, I used induction on n, for $n=1$, comparing degree of the factors we get the answer and for higher degrees we use that $x$ is a factor of $ax^n$ so, x must divide one of the factors of $ax^n$, more precisely, if $fg=ax^n$ then $x$ divides at least one of $f$ and $g$.
I am stuck on how to deal with it for multivariable case.
Suppose $f$ and $g$ are elements of a graded domain such that $fg$ is homogeneous. Then $fg={f_d}{g_{d'}}$, where $d$ and $d'$ are the degrees of $f$ and $g$ respectively. This implies that all the other components of $fg$ vanish. Suppose $l$ and $l'$ are degrees of the non-zero components of least degree of $f$ and $g$ respectively. If either $l<d$ or $l'<d'$, $f_lg_{l'}=0$ and this implies that $f_l=0$ or $g_{l'}=0$ contradicting our assumption that $f_l$ and $g_l'$ are non-zero. So, $l=d$ ad $l'=d'$ and therefore, $f$ and $g$ are homogeneous.