Recently, there have been attempts to integrate various speech processing tasks into a unified model. However, few previous works directly demonstrated that joint optimization of diverse tasks in multitask speech models has positive influence on the performance of individual tasks. In this paper we present a multitask speech model -- PolySpeech, which supports speech recognition, speech synthesis, and two speech classification tasks. PolySpeech takes multi-modal language model as its core structure and uses semantic representations as speech inputs. We introduce semantic speech embedding tokenization and speech reconstruction methods to PolySpeech, enabling efficient generation of high-quality speech for any given speaker. PolySpeech shows competitiveness across various tasks compared to single-task models. In our experiments, multitask optimization achieves performance comparable to single-task optimization and is especially beneficial for specific tasks.