In this paper, we put forth a combined method for calculation of all real zeroes of a polynomial
equation through the Adomian decomposition method equipped with a number of developed
theorems from matrix algebra. These auxiliary theorems are associated with eigenvalues of
matrices and enable convergence of the Adomian decomposition method toward different real roots of the target polynomial equation. To further improve the computational speed of our technique, a nonlinear convergence accelerator known as the Shanks transform has optionally been employed. For the sake of illustration, a number of numerical examples are given.