Why would I want to multiply two polynomials? I'm hoping that this isn't such a basic question that it gets completely laughed off the site, but why would I want to multiply two polynomials together? 
I flipped through some algebra books and have googled around a bit, and whenever they introduce polynomial multiplication they just say 'Suppose you have two polynomials you wish to multiply', or sometimes it's just as simple as 'find the product'. I even looked for some example story problems, hoping that might let me in on the secret, but no dice. 
I understand that a polynomial is basically a set of numbers (or, if you'd rather, a mapping of one set of numbers to another), or, in another way of thinking about it, two polynomials are functions, and the product of the two functions is a new function that lets you apply the function once, provided you were planning on applying the original functions to the number and then multiplying the result together. 
Elementary multiplication can be described as 'add $X$ to itself $Y$ times', where $Y$ is a nice integer number of times. When $Y$ is not a whole number, it doesn't seem to make as much sense.
Any ideas?

When you take calculus, you will need to factor a polynomial p as a product of two polynomials a and b.  If you know how polynomial multiplication works, then finding factorizations is easier.  Learn how to multiply now so that you can factor easily later. :)
