The authors discuss an unconstrained minimization algorithm in which steps are generated by augmenting the quadratic Newton model by low-rank third- and fourth- order terms. These additional tensor terms are chosen to make the model function interpolate function and derivative information at previous iterates. The Newton model is not completely discarded; both Newton and tensor model steps are calculated, and the algorithm chooses the one that gives the better reduction in function value. A trust region technique is used for robustness. The overall algorithm performs consistently better than Newton's method on a suite of standard test problems, particularly when the Hessian is singular at the solution.