Suppose $C$ is an affine plane curve and $L$ is a line in $A^2(k)$, $L \not\subset C$. Suppose $C=V(F), F \in K[X,Y]$ a polynomial of degree $n$. Show that $ L \cap C$ is a finite set of no more than $n$ points.
I started like this: Suppose $L=V(Y-(aX+b))$. Then Considering $F(X,aX+b) \in k[X]$.
Then I am stuck.
A point $(x,y)$ in $L\cap C$ satisfies $y=ax+b$ and $F((x,y))=0$. So every such point is a zero of the polynomial $F(X,aX+b)$. Inductively it is easy to show that $F(X,aX+b)$ defines a polynomial of degree $\leq n$ in $K[X]$. So there are at most $n$ points in $L\cap C$.
EDIT: obviously, I assume $C\neq L$.