We　propose　a　polynomial　expansion　method　to　investigate　periodic　motions　around　both　collinear　and　triangular　libration　points　in　the　planar　circular　restricted　three-body　problem.　The　polynomial　expansion　method　focuses　on　the　approximate　nonlinear　polynomial　relations　between　the　two　directions　in　the　motion　plane,　which　provide　an　alternative　way　to　inspect　the　periodic　motions.　Based　on　the　nonlinear　polynomial　relations,　the　planar　two-dimensional　system　is　decoupled.　As　an　example,　the　order　analytical　solutions　for　the　periodic　motion　are　determined　by　solving　the　decoupled　system.　The　efficiency　of　the　polynomial　expansion　method　has　been　validated　by　numerical　method.