$p \in C - D$, inflection point for $C$ iff inflection point for $C \cup D$.

Question

Show that if $C$ and $D$ are projective curves in $\mathbb{P}_2$ and $p \in C - D$ then $p$ is a point of inflection for the curve $C$ if and only if $p$ is a point of inflection for the curve $C \cup D$.

Answer 1

Personally I think Mostafa's answer in the comments is the best, but mine is more algebraic.

A point $p$ on an algebraic curve $C$ given by a homogenous polynomial $f$ is an inflection point iff the Hessian $$ \mathcal{H} = \text{det}\bigg{(}\frac{\partial^2f}{\partial_{x_i}\partial_{x_j}}\bigg{)} $$ vanishes at the point $p$. Let $D$ be given by a homogenous polynomial $g$. Then if one were so inclined to apply the chain rule twice to get the Hessian of the polynomial $fg$, which defines $C \cup D$, one could show that the point $p$ is an inflection point of $C \cup D$.