In this paper, researched by A. T. Fuller (deceased) and written by E. I. Jury, the necessary and sufficient conditions for aperiodicity which require the roots of a real polynomial to be real, simple, and confined to a specified interval on the real axis are presented in terms of four theorems. These theorems require a positivity test of a certain auxiliary polynomial in the specified interval on the real axis. One of these theorems is a generalization of an earlier published paper by Meerov and Jury (1998). An application of these theorems occurs when the real roots are to be confined to the interval (0,1), a condition related to discrete-time aperiodicity. Another application is related to stable aperiodic continuous-time systems where the real roots are to be confined (-infinity, 0). An example is presented to indicate the application for the discrete-time case. This paper also represents a generalization of Lipka Theorem (Lipka 1943).