Convolutional neural networks (CNNs) tend to look at methods to improve performance by adding more input data, making modifications on existing data, or changing the design of the network to better suit the variables. The goal of this work is to supplement the small number of existing methods that do not use one of the previously mentioned techniques. This research aims to show that with a standard CNN, classification accuracy rates have the potential to be improved without changes to the data or major network design modifications, such as adding convolution or pooling layers. A new layer is proposed that will be inserted in a similar location as the non-linearity functions in standard CNNs. This new layer creates a localized connectivity for each perceptron to a polynomial of N^th degree. This can be performed in both the convolutional portion and the fully connected portion of the network. The proposed polynomial layer is added with the idea that the higher dimensionality enables a better description of the input space, which leads to a higher classification rate. Two different datasets, MNIST and CIFAR10 are utilized for classification, each containing 10 distinct classes and having similar training set sizes, 60,000 and 50,000 images, respectively. These datasets differ in that the images are 28 × 28 grayscale and 32 × 32 RGB, respectively. It is shown that the added polynomial layers enable the chosen CNN design to have a higher rate of accuracy on the MNIST dataset. This effect was only similar at a lower learning rate with the CIFAR10 dataset.