Bicoprime factorizations (BCFs) are a generalization of the well known coprime factorizations commonly used in control theory. However they have received negligible attention from the academic community so far. This technical note lays the foundations of a BCF theory. The theory is built from the ground up, starting with the basic characteristics of such factorizations before moving on to state space parameterizations of BCFs and internal stability. Some advantages of BCFs are outlined including the possibility of reduced dimension internal stability tests. An uncertainty structure induced by BCFs is also examined and the associated robust stability analysis tests provided. In multiple instances it is shown how coprime factor results have their roots in the more abstract, and more general, BCFs.