In this paper, we use the idea of normal family to investigate the uniqueness problems of meromorphic functions when certain non-linear differential polynomial sharing a non-zero polynomial with certain degree. We obtain some results which will not only rectify the recent results of P. Sahoo and H. Karmakar but also improve and generalize some recent results of L. Liu, H. Y. Xu, T. B. Cao and S. Liu and P. Sahoo and H. Karmakar in a large extent.