In this work, a new method is introduced for solving large polynomial systems for the kinematic synthesis of linkages. The method is designed for solving systems with degrees beyond 100,000, which often are found to possess quantities of finite roots that are orders of magnitude smaller. Current root-finding methods for large polynomial systems discover both finite and infinite roots, although only finite roots have meaning for engineering purposes. Our method demonstrates how all infinite roots can be precluded in order to obtain substantial computational savings. Infinite roots are avoided by generating random linkage dimensions to construct startpoints and start systems for homotopy continuation paths. The method is benchmarked with a four-bar path synthesis problem.