This paper presents a novel distributed computing framework, DCbMF (Distributed Computing by Matrix Factorization), for the efficient computation of linearly separable functions. Our framework operates within a multicast network of interconnected computing nodes to compute a set of output functions from input data through an efficient three-step process - Map, Shuffle, and Reduce. We cast the computation procedure as a sparse matrix factorization problem to achieve efficient communication and computation. To this end, we formulate an ℓ0 optimization problem that seeks to minimize the number of nonzero elements in each matrix factor. Due to the intractability of ℓ0 minimization, we turn to a relaxed ℓ1 formulation of the problem. We devise a modified alternating direction method of multipliers to solve the biconvex optimization problem and prove the convergence of our algorithm to a stationary solution. The numerical experiments show that DCbMF outperforms similar computing frameworks for linearly separable function computation, achieving a substantial reduction in communication overhead by 98% while maintaining the same computation cost. Notably, leveraging sparse matrix factorization and alternating optimization highlights a fundamental tradeoff between compu-tation and communication costs and paves the way for scalable and efficient distributed computing applications.