In this work we advance the understanding of the fundamental limits of computation for Binary Polynomial Optimization (BPO), which is the problem of maximizing a given polynomial function over all binary points. In our main result we provide a novel class of BPO that can be solved efficiently both from a theoretical and computational perspective. In fact, we give a strongly polynomial-time algorithm for instances whose corresponding hypergraph is beta-acyclic. We note that the beta-acyclicity assumption is natural in several applications including relational database schemes and the lifted multicut problem on trees. Due to the novelty of our proving technique, we obtain an algorithm which is interesting also from a practical viewpoint. This is because our algorithm is very simple to implement and the running time is a polynomial of very low degree in the number of nodes and edges of the hypergraph. Our result completely settles the computational complexity of BPO over acyclic hypergraphs, since the problem is NP-hard on alpha-acyclic instances. Our algorithm can also be applied to any general BPO problem that contains beta-cycles. For these problems, the algorithm returns a smaller instance together with a rule to extend any optimal solution of the smaller instance to an optimal solution of the original instance.