OICC Press Mathematical Analysis and its Contemporary Applications 4 1 2022 01 01 On the zeros and critical points of a polynomial 10.30495/maca.2021.1938758.1028 EN Mohammad Ibrahim Mir Department of Mathematics, University of Kashmir, South Campus, Anantnag 192101, Jammu and Kashmir, India Irfan Ahmad Wani Department of Mathematics, University of Kashmir, South Campus, Anantnag 192101, Jammu and Kashmir, India Ishfaq Nazir Department of Mathematics, University of Kashmir, South Campus, Anantnag 192101, Jammu and Kashmir, India Journal Article 2022 01 01 Let P(z) = a0 + a1z + ... + an-1z^{n-1} + z^n be a polynomial of degree n. The Gauss-Lucas Theorem asserts that the zeros of the derivative P'(z) = a1 + ... + (n - 1)an-1z^{n-2} + nz^{n-1}, lie in the convex hull of the zeros of P(z). Given a zero of P(z) or P'(z), A. Aziz [1], determined regions which contain at least one zero of P(z) or P'(z) respectively. In this paper, we give simple proofs and improved version of various results proved in [1], concerning the zeros of a polynomial and its derivative.