Traditional drug experiments to find synergistic drug pairs are time-consuming and expensive due to the numerous possible combinations of drugs that have to be examined. Thus, computational methods that can give suggestions for synergistic drug investigations are of great interest. Here, we propose a Non-negative Matrix Tri-Factorization (NMTF) based approach that leverages the integration of different data types for predicting synergistic drug pairs in multiple specific cell lines. Our computational framework relies on a network-based representation of available data about drug synergism, which also allows integrating genomic information about cell lines. We computationally evaluate the performances of our method in finding missing relationships between synergistic drug pairs and cell lines, and in computing synergy scores between drug pairs in a specific cell line, as well as we estimate the benefit of adding cell line genomic data to the network. Our approach obtains very good performance (Average Precision Score equal to 0.937, Pearson's correlation coefficient equal to 0.760) when cell line genomic data and rich data about synergistic drugs in a cell line are considered. Finally, we systematically searched our top-scored predictions in the available literature and in the NCI ALMANAC, a well-known database of drug combination experiments, proving the goodness of our findings.