A quadrature free discontinuous Galerkin method using a general polynomial basis set is developed. It is shown that this method allows for the use of orthogonal and near orthogonal polynomials, whose properties not only lead to sparse elemental matrices, but also reduce the computational effort required in the evaluation of elemental boundary integrals. Expansion of the nonlinear flux expressions is accomplished easily using nodal polynomials such as Lagrange polynomials. Since most of the work involved in a discontinuous Galerkin solver is concentrated in the calculation of the boundary integrals involving coordinate transformations and nonlinear flux expansions, the present approach proves to be an efficient one. The method is applied to solve benchmark problems in aeroacoustics, involving uniform and non-uniform mean flows in two dimensions. Basis sets consisting of polynomials of up to 5th degree are used on an unstructured grid, and a study of both h and p refinement is presented. The performance of the method is compared to the quadrature free method using moment polynomials.