Definitions
Any mathematical discussion starts off with some definitions. Mathematics is a very precise subject so mathematicians try to be very precise with their terms. This can be a source of confusion to students since the results often seem wordy and convoluted. Throughout this class, we'll try to work through that by giving the mathematical definition along with an English translation and lots of examples.
A polynomial is an expression involving the sum of products of constants and variables raised to integer powers. An individual part of the sum is called a term. The constants by which terms are multiplied are called coefficients.
Let's take that a piece at a time. A polynomial starts off as the product of some variables such as xy and x. Next we're allowed to raise the variables to integer powers like x^{2}y and x^{2}. We can also multiply those items by numbers: 2x^{2}y and x^{2}. Finally, we can add them together to get 2x^{2}y + x^{2}. That gives us a basic polynomial where 2x^{2}y and x^{2} are the terms and 2 and 1 are the coefficients.
You might be wondering where I got the 1 from as a coefficient in my example. Remember that 1 times a number (or a variable) is just the number so when there's a 1 in front of a variable, we usually don't write it. In this case, you could "fully" write out the polynomial as 2x^{2}y + 1x^{2} if you wanted to make the coefficient explicit.
Here are some more examples of things that are polynomials and things that aren't.

I think it's easiest to look at an expression and decide if it isn't a polynomial. Here are some things to look for:
 It isn't a polynomial if it has variables in the denominator.
 It isn't a polynomial if it has exponents that aren't positive integers.
 It isn't a polynomial if it has any special functions like logarithms or trig functions.
 Be careful simplifying first  especially if you're getting a variable out of the denominator.
Is the following expression a polynomial?