The　harmonic　balance　coupled　with　the　continuation　algorithm　is　a　well-known　technique　to　follow　the　periodic　response　of　dynamical　system　when　a　control　parameter　is　varied.　However,　deriving　the　algebraic　system　containing　the　Fourier　coefficients　can　be　a　highly　cumbersome　procedure,　therefore　this　paper　introduces　polynomial　homotopy　continuation　technique　to　investigate　the　steady　state　bifurcation　of　a　two-degree-of-freedom　system　including　quadratic　and　cubic　nonlinearities　subjected　to　external　and　parametric　excitation　forces　under　a　nonlinear　absorber.　The　fractional　derivative　damping　is　considered　to　examine　the　effects　of　different　fractional　order,　linear　and　nonlinear　damping　coefficients　on　the　steady　response.　By　means　of　polynomial　homotopy　continuation,　all　the　possible　steady　state　solutions　are　derived　analytically,　i.e.　without　numerical　integration.　Coexisting　periodic　solutions,　saddle-node　bifurcation　and　various　effects　of　fractional　damping　on　the　steady　state　response　are　found　and　investigated.　It　is　shown　that　the　fractional　derivative　order　and　damping　coefficient　change　the　bifurcating　curves　qualitatively　and　eliminate　the　saddle-node　bifurcation　during　resonance.　Moreover,　the　system　response　depicts　bigger　and　bigger　region　of　hard-spring　bistability　with　increasing　fractional　derivative　order,　but　the　region　of　hard-spring　bistability　of　steady　response　becomes　gradually　small　and　then　disappears　when　we　increase　the　linear　and　nonlinear　damping　coefficients.　In　addition,　the　analytical　results　are　verified　by　comparison　with　the　numerical　integration　ones,　it　can　be　found　that　the　present　approximate　resonance　responses　are　in　good　agreement　with　numerical　ones.