An important factorization algorithm for polynomials over finite fields was developed by Niederreiter. The factorization problem is reduced to solving a linear system over the finite field in question, and the solutions are used to produce the complete factorization of the polynomial into irreducibles. One charactersistic feature of the linear system arising in the Niederreiter algorithm is the fact that, if the polynomial to be factorized is sparse, then so is the Niederreiter matrix associated with it. In this paper, we investigate the special case of factoring trinomials over the binary field. We develop a new algorithm for solving the linear system using sparse Gaussian elmination with the Markowitz ordering strategy. Implementing the new algorithm to solve the Niederreiter linear system for trinomials over F2 suggests that, the system is not only initially sparse, but also preserves its sparsity throughout the Gaussian elimination phase. When used with other methods for extracting the irreducible factors using a basis for the solution set, the resulting algorithm provides a more memory efficient and sometimes faster sequential alternative for achieving high degree trinomial factorizations over F2.