The aim of this dissertation is to develop an extension of the Taylor theorem related to linear differential operators with constant coefficients. For this aim, we employ divided differences with repeated arguments to describe the characteristic element from the kernel of the differential operator. We establish the extension of the Taylor theorem concerning exponential polynomials and investigate its implications, incorporating integral remainder terms and mean value type theorems. Furthermore, we establish various factorization results and utilize them to derive estimates for linear functionals through a generalized Taylor theorem. Moreover, we establish several error bounds, including their applications to the trapezoidal rule and a Simpson formula rule.