(eng): The paper deals with solving algebraic Riccati equations with two-sided polynomials, which arise in some applications of optimal control of linear time-invariant spatially distributed systems. The conditions are given for the existence of a solution in the set of finite-order polynomials. If such a solution does not exist, the infinite-order solution is truncated and given in the form of polynomial of an arbitrary high order. The proposed numerical algorithm is based on the discrete Fourier transform theory. An example on optimal control of spatially-distributed systems is also given. The proposed algorithm is used to design of the distributed LQ controller.