I am looking for the definition of a polynomial.

In case you had trouble understanding those two complex definitions, here's a simpler one.

First, poly means many, and nomial means name.  So, a polynomial has more than one part to it.  

It is a mathematical expression -- in other words, a string of mathematical symbols that aren't nonsense.

There are three kinds of things that can be in a polynomial:

1. variables (like x)
2. constants (like the 3 in 3x, or the 4 in x^2 + 2x + 4)
3. whole number exponents that aren't negative (like x^2, but not x^-2 or x^1/4)

Lastly, the expression can have addition, subtraction and multiplication, but not division.

I hope this is helpful.