AbstractLet be the matrix algebra over and be the invertible elements in . Inspired by Kaplansky–Lv́ov conjecture, we explore the image of polynomials with constants, namely polynomials from the free algebra . In this article, we compute the images of the polynomial maps given by (a) generalized sum of powers and (b) generalized commutator map , where , are nonzero elements of when is an algebraically closed field. We show that the images of these maps are vector spaces. For the polynomial in (a), we compute the images by fixing a simultaneous conjugate pair for , and it turns out that it is surjective in most cases.