A new recursion in only one variable allows very simple verifications of Bressoud's polynomial identities, which lead to the Rogers-Ramanujan identities. This approach might be compared with an earlier approach due to Chapman. Applying the $q$-Chu-Vandermonde convolution, as suggested by Cigler, makes the computations particularly simple and elementary. The same treatment is also applied to the Santos polynomials and perhaps more polynomials from a list of Rogers-Ramanujan like polynomials \cite{Sills}.