First-order term unification is an essential concept in areas like functional and logic programming, automated deduction, deductive databases, artificial intelligence, information retrieval, compiler design, etc. We build upon recent developments in grammar-based compression mechanisms for terms and investigate algorithms for first-order unification and matching on compressed terms. We prove that the first-order unification of compressed terms is decidable in polynomial time, and also that a compressed representation of the most general unifier can be computed in polynomial time. Furthermore, we present a polynomial time algorithm for first-order matching on compressed terms. Both algorithms represent an improvement in time complexity over previous results [GGSS09, GGSS08]. We use several known results on the tree grammars used for compression, called singleton tree grammars (STG)s, like polynomial time computability of several subalgorithmms: certain grammar extensions, deciding equality of represented terms, and generating their preorder traversal. An innovation is a specialized depth of an STG that shows that unifiers can be represented in polynomial space