In Hartshorne's "Algebraic Geometry" there is this nice exercise I.7.5:
If $Y$ is an irreducible plane curve of degree $d>1$ having a point of multiplicity $d-1$ then $Y$ is a rational curve.
I wonder if the converse is true i.e. if I have a parameterizable/rational curve of degree $d$ can it happen that at all points of the curve the multiplicity is not $d-1$?