Public vs. private universities is a debatable issue, and it creates polarization on social media in Bangladesh. Debate on quality, jobs, and prestige is passionate among the students, parents, and graduates, the majority of whom speak Bengali, a low-resource language. To measure this polarization, this paper introduces a manually annotated dataset of 4,060 Bengali comments labeled as Pro-Public, Pro-Private, or Neutral. We evaluated the quality of our annotations by Fleiss's Kappa agreement that was 0.89, corresponding to a high agreement among annotators. The classical ML (SVM, Random Forest, XGBoost), BiLSTM network, hybrid BanglaBERT+XGBoost models and the state-of-the-art zero-shot LLMs (Claude Sonnet 4, DeepSeek-V3.1, Llama 4 Maverick, Kimi K2 Thinking, Qwen3-235B Thinking) models are evaluated. The accuracy of BanglaBERT+XGBoost is 91.81% and macro F1 score is 91.70%, which is higher than all the supervised baselines. The zero-shot Llama 4 Maverick Thinking achieves a macro F1 of 0.931 (overall accuracy of 93.31%) without any fine-tuning. All machine learning (ML), deep machine learning (DL) and transformer models were outperformed by the zero-shot Llama 4 Maverick model. Polarization also is evident, in some ways more clearly in the Pro-Private comments, which emphasize modern facilities and timely graduation, versus the Pro-Public comments, which emphasize affordability and government jobs. Our findings open new directions for analyzing social media polarization in low-resource languages.