In this paper, we investigate the role of cell death in promoting pattern formation within bacterial biofilms. To do this we utilise an extension of the model proposed by Dockery and Klapper [13], and consider the effects of two distinct death rates. Equations describing the evolution of a moving biofilm interface are derived, and properties of steady state solutions are examined. In particular, a comparison of the planar behaviour of the biofilm interface in the different cases of cell death is investigated. Linear stability analysis is carried out at steady state solutions of the interface, and it is shown that, under certain conditions, instabilities may arise. Analysis determines that, while the emergence of patterns is a possibility in 'deep' biofilms, it is unlikely that pattern formation will arise in 'shallow' biofilms.