Math News: Prime Numbers

Introduction

Every Algebra student learns about polynomials. These mathematical expressions are ones constructed using variables and constants along with the operations of addition, subtraction, multiplication and non-negative positive powers.
A simple example of a polynomial in X is:

3X2+4X+3

The number of terms of the polynomial above is three (3). Another definition of a polynomial is that is the sum of one or more non-zero terms. The number of terms is always a finite number.
Another example of a polynomial is:

A2+2AB+B2

This polynomial is the result of the expansion of the following expression:

(A+B)(A+B) or
(A+B)2

It can be observed that the number of terms of the polynomial A+B is 2.

When said polynomial is raised to the 2nd power and expanded, the final number of terms is 3.

It is logical then that one would be curious to know the final number of terms when raising A+B to the 3rd or 4th or nth powers.

What follows is a series of observations about polynomials, numbers and mathematics in general, made during my effort to answer the question “How many terms does a polynomial of any number of terms raised to any power have after it is completely expanded?”

Not all topics are thoroughly studied since there are infinite paths and forks in the paths that without a doubt would lead to interesting destinations, I have identified some of these with the label “Additional Research #”. I intend to come back to these points.

I am sure that much if not all of what follows has been known to others far more illustrious, but it is my sincere hope that it is at least entertaining and interesting to anyone that loves mathematics.