Projection-based model-order reduction (PMOR) aims to reduce the computational cost associated with many-query tasks involving high-dimensional dynamical models. This is typically achieved by projecting the full-order state onto a low-dimensional trial basis and then projecting the governing equations onto an appropriate test basis to obtain a reduced-order model. In this work, we focus on dynamical systems with polynomial nonlinearity, solved using implicit time integration schemes via Newton solvers. For such time integration schemes, PMOR commonly involves projecting the Newton solver’s residual and Jacobian onto a test basis [1]. However, when the dynamical system includes nonlinear terms, evaluating these projected terms often exhibits an operation count complexity that scales directly with the dimension of the full-order model, thereby diminishing computational cost savings. To address this issue, hyper-reduction techniques are typically used to approximate nonlinear terms via sparse sampling, enabling efficient evaluation of their projected contributions. Although hyper-reduction schemes have been demonstrated to efficiently and accurately solve nonlinear systems [2], they are ultimately an approximation of the nonlinear term. Such an approximation introduces both an additional hyperparameter and additional sources of error. The current work aims to overcome this limitation by investigating a class of nonlinear dynamical systems of polynomial form for which the projected form of the Newton solver’s residual and Jacobian can be written explicitly in a low-dimensional form, thereby bypassing the need for a hyper-reduction scheme. We demonstrate the efficacy of this class of hyper-reduction-free (HRF) approaches for both Galerkin and Petrov-Galerkin projections, highlighting their versatility. Numerical experiments compare this class of HRF approaches with hyper-reduction-based approaches in terms of both accuracy and computational cost. [1] K. Carlberg, C. Bou-Mosleh, C. Farhat, Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations, International Journal for Numerical Methods in Engineering, 86, pp. 155-181, 2011. [2] C. Farhat, T. Chapman, P. Avery, Structure-preserving, stability, and accuracy properties of the energy-conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models, 86, pp. 1077-1110, 2015.