The field of tropical algebra is closely linked with the domain of neural networks with piecewise linear activations, since their output can be described via tropical polynomials in the max-plus semiring. In this work, we attempt to make use of methods stemming from a form of approximate division of such polynomials, which relies on the approximation of their Newton Polytopes, in order to minimize networks trained for multiclass classification problems. We make theoretical contributions in this domain, by proposing and analyzing methods which seek to reduce the size of such networks. In addition, we make experimental evaluations on the MNIST and Fashion-MNIST datasets, with our results demonstrating a significant reduction in network size, while retaining adequate performance.