In this talk, we will shortly present a direct and reverse way to develop the polynomial method that relies on the Combinatorial Nullstellensatz. The direct and usual way states that a multivariate polynomial of small degree cannot vanish on a large cartesian product provided that a specified coefficient is non zero. The reverse way relies on the coefficient formula and establishes an expression for this specified coefficient. This double interpretation of the polynomial method allows to shorten the proofs of the Cauchy-Davenport and the Dias da Silva-Hamidoune theorems and a new result on the cardinality of sets of subsums. Moreover tese proofs do not require any computations and do imply the critical cases of these three problems: arithmetical progressions.