I have a homework question that asks
Let $f$ and $g$ be two quadratic polynomials (with coefficients in $\mathbb{C}$) that share no common linear factors, and let $C_f$ and $C_g$ be the zero set of $f$ and $g$, respectively. Let $p$ and $q$ be distinct points in $C_f \cap C_g$, and let $L$ be the line through $p$ and $q$. Show that there are constants $c_1$ and $c_2$ (not both zero) such that $c_1f + c_2g$ is identically zero on $L$ and is the product of linear polynomials.
This problem has been confusing me for some time. Here's where I'm currently at with it. Since $f$ and $g$ have degree $2$, I can say that $C_f = \{ p_1, p_2 \}$ and $C_g = \{ q_1, q_2 \}$ for some points $p_i$ and $q_i$. Since $p, q \in C_f \cap C_g$ are distinct, can I not then conclude that $C_f = C_g = \{ p, q \}$? But now $f$ and $g$ share a common linear factor (actually it looks like they are associate). What am I doing wrong?
