The powerful text understanding and generation capabilities of large language models (LLMs) have brought new vitality to general recommendation with implicit feedback. One possible strategy involves generating a unique user (or item) profile from historical interaction data, which is then mapped to a semantic representation in the language space. However, a single-instance profile may be insufficient to comprehensively capture the complex intentions behind a user's interacted items. Moreover, due to the inherent instability of LLMs, a biased or misinterpreted profile could even undermine the original recommendation performance. Consequently, an intuitive solution is to generate multiple profiles for each user (or item), each reflecting a distinct aspect of their characteristics. In light of this, we propose a unified recommendation framework with multi-faceted profile extrapolation (ProEx) in this paper. By leveraging chain-of-thought reasoning, we construct multiple distinct profiles for each user and item. These new profiles are subsequently mapped into semantic vectors, extrapolating from the position of the original profile to explore a broader region of the language space. Subsequently, we introduce the concept of environments, where each environment represents a possible linear combination of all profiles. The differences across environments are minimized to reveal the inherent invariance of user preferences. We apply ProEx to three discriminative methods and three generative methods, and conduct extensive experiments on three datasets. The experimental results demonstrate that ProEx significantly enhances the performance of these base recommendation models.