I came across this question in Algebraic Geometry: A Problem Solving Approach: Let P(x,y,z) be an irreducible homogeneous second degree polynomial. Show that the intersection multiplicity of V(P) with any line l is at most 2.
I tried writing down the general form of a second degree polynomial and taking second order partials and showed that no point can have intersection multiplicity more than 2 this way. However, I did not use the fact that P was irreducible at any point and presume there was some error I made. Any advice into the error and how to solve it would be greatly appreciated.
Also, I'm just starting to teach myself algebraic geometry so I don't yet have much familiarity with the more abstract concepts.