  We consider the Cauchy problem for an equation of the form
\partial_t+\partial_x^3)u=F(u,u_x,u_{xx}) where F is a polynomial with no
constant or linear terms and no quadratic uu_{xx} term. For a polynomial
nonlinearity with no quadratic terms, Kenig-Ponce-Vega proved local
well-posedness in H^s for large s. In this paper we prove local well-posedness
in low regularity Sobolev spaces and extend the result to certain quadratic
nonlinearities. The result is based on spaces and estimates similar to those
used by Marzuola-Metcalfe-Tataru for quasilinear Schrodinger equations.
