Question about intersection multiplicity of a curve and it's tangent line If we have a double point $a$ on some complex curve, call it $C$, defined by some polynomial $f$ and we have only one tangent line at $a$, call it $T_l$, then the intersection multiplicity $I(a,f \cap T_l) =3$. 
I think this is a simple application of Bézout's theorem, right? Because every point on a line is non-singular $a$ has multiplicity 1 thus when we and the two multiplicities we get 3 and we are done. 
Is this correct thinking on my part or is there a whole different approach to this problem? Thank you guys!

This is not true, unless $C$ is a cubic curve.
For example, take the curve with equation $$(x^2+y^2-3x)^2 -4x^2(2-x)=0$$
Notice that the tangent line at the origin is $x=0$, and its intersection multiplicity with $C$ is $4$.
