Difference between optimizing single polynomial neuron vs multiple perceptron? Is there any specific reason why we usually prefer to use a neural network with, let's say, 6 hidden neurons rather than one single neuron representing the following function?
$$ax^3+bx^2+cx+d$$
Is it harder to optimize the latter one?
 A: It's mainly convention to think of a neuron as a nonlinear activation applied on a linear input, e.g. $f(ax_1+bx_2+c)$. The point is that after a few layers of connections, you'll have reasonable approximations to stuff like $x^2,x^3$ if necessary. The allowed modifications to this become whether you wanna use ReLu, tanh, etc. activations.  It's also "minimal" in that linear combinations are fairly cheap to compute, followed by nonlinear activation. This speeds up individual neuron forward and backward passes (but can arguably require a slightly deeper network). Overall, speed is still an active research area.
It also makes input normalization less of a headache. $x^3$ can grow out of control with a large input, or look abnormally small for a small input. Whereas with a linear input, you can always control for the magnitude and range of $ax_1+bx_2+c$ via a constant shift. You could definitely get this to work for $x^3$ as well, but then it becomes more annoying to to debug unless all your neurons follow the same conventional form of input.
