Polynomials and integers have many similarities, but when it comes to dividing, they each go their own way. You might even say they "divide." Ooh, see what we did there?
If we add any two integers, subtract one integer from another, or multiply two integers, we always find an answer that's also an integer. When we divide one integer by another, however, we don't necessarily have an integer. We might find a rational number. It would be nice if division followed the example set by addition, subtraction, and multiplication instead of being a pain in the hiney, but there's always one...
It's the same deal with polynomials. If we add, subtract, or multiply two polynomials, we get an answer that's also a polynomial. When we try to divide one polynomial by another, however, the answer won't necessarily be a polynomial. For example, "1" is a polynomial, and "x" is a polynomial, but 1/x is not a polynomial. This is an example of a rational function, which we'll talk more about in the next unit. Sorry, we don't mean to be such a tease.