Why can there be radicals, negative exponents, or fractional exponents in a polynomial?
This was intriguing. With a bit of wikipedia-ing: 1. When you DO have radicals, negative exponents, or fractional exponents, the expression is a rational expression, i.e. a polynomial divided by another polynomial. 2. The derivative and antiderivative of a polynomial expression needs to also be polynomial expressions. An easy example, therefore, is 1/x, whose antiderivative is ln(x) and definitely not a polynomial. 3. By being restrictive, you get more information. For example, according to the fundamental theorem of algebra, every polynomial has a root, if we count the complex numbers. If you let in things like 1/x, then the expression does not necessarily have a root, and our assumptions are much weaker.
