It　is　known　that　computing　the　largest　(smallest)　Z-eigenvalue　of　a　symmetric　tensor　is　equivalent　to　maximizing　(minimizing)　a　homogenous　polynomial　over　the　unit　sphere.　Based　on　such　a　reformulation,　we　shall　propose　a　feasible　trust-region　method　for　calculating　extreme　Z-eigenvalues　of　symmetric　tensors.　One　basic　feature　of　the　method　is　that　the　true　Hessian,　which　is　ready　for　polynomials,　is　utilized　in　the　trust-region　subproblem　so　that　any　cluster　point　of　the　iterations　can　be　shown　to　satisfy　the　second-order　necessary　conditions.　The　other　feature　is　that　after　a　trial　step　d(k)　is　provided　by　solving　the　trust-region　subproblem　at　the　current　point　x(k),　the　projection　of　x(k)　+　d(k)　to　the　unit　sphere,　instead　of　the　point　x(k)　+　d(k)　itself,　is　judged　and　if　successful,　is　used　for　the　next　point.　Global　convergence　and　local　quadratic　convergence　of　the　feasible　trust-region　method　are　established　for　the　tensor　Z-eigenvalue　problem.　The　preliminary　numerical　results　over　several　testing　problems　show　that　the　feasible　trust-region　method　is　quite　promising.