Suppose that for a homogeneous polynomial $f(x,y,z)$ we have $f(x,y,z)=g(x,y,z) \cdot h(x,y,z)$. Assume $f_d$ and $g_d$ are the dual curves of $f$ and $g$ respectively. Is it true that $g_d$ is a factor of $f_d$?
My guess is that it is, since if a point $(X,Y,Z)$ is in the dual curve of $g$, then it must satisfy $(X,Y,Z)= \mu \nabla g(p,q,r)$ for some point $(p,q,r)$ such that $g(p,q,r)=0$ and some constant $\mu$. But then, $(X,Y,Z)$ also satisfies $(X,Y,Z)= \lambda \nabla f(p,q,r)$, since $$ \nabla f(p,q,r)= g(p,q,r) \nabla h(p,q,r) + h(p,q,r) \nabla g(p,q,r) = h(p,q,r) \nabla g(p,q,r) $$ and hence $(X,Y,Z)$ is a multiple of $\nabla f(p,q,r)$ (as long as $h(p,q,r)\neq 0$, which I think we can always assume). Is this correct?