<p>This paper discloses the unique dynamic behavior and vibration mitigation performance of a polynomial nonlinear vibration absorber (PNVA), a single-degree-of-freedom nonlinear vibration absorber with positive linear and cubic stiffnesses and negative quadratic stiffness, attached to a linear primary system subjected to harmonic excitation. The slow flow equations of the compound system are derived analytically using the harmonic balance method, based on which the analyses of bifurcation, stability and frequency response are carried out. The results show that the primary system coupled with a PNVA has different dynamic behavior compared to that coupled with a traditional nonlinear energy sink (NES), such as the existence of at most five periodic solutions to the slow flow equations, the possible existence of two closed regions enclosed by boundaries for Hopf bifurcations, and the possible existence of two folds in a slow invariant manifold curve. Numerical simulations are performed to validate theoretical analyses, explore comprehensive dynamics, and optimize the design of a PNVA. It is shown that the numerical and analytical results of steady-state periodic response present a good agreement with each other. The mass ratio and damping ratio of a PNVA, initial conditions, and external excitations all have effects on the response regimes of the compound system such that it may undergo a desirable low-amplitude periodic/quasi-periodic response, strongly modulated response (SMR) or an undesirable large-amplitude periodic response. Comparison among the vibration absorbers attached to a primary system with stiffness uncertainty shows that the optimal PNVA has better vibration mitigation efficiency and robustness against frequency variations than the optimal tuned mass damper and Cubic NES, and the optimal PNVA has better robustness against variations in external excitation amplitude than the optimal Cubic NES.</p>