Wikipedia says, "In mathematics , a polynomial is an expression consisting of variables and coefficients which only employs the operations of addition , subtraction , multiplication , and non-negative integer exponents."
My question is why can't variables in polynomials can't take any fraction or negative exponents? Is it just that a collection of algebraic expressions are classified this way, or it has some mathematical reason behind it?
That is just the definition of a polynomial. You can of course have fractions, leading you to the so called rational functions. The point is just that polynomials arise in many cases and thus it is important to study them. If you allow for fractions you will gain more elements, yes, but you will also loose some structure that you might get helpful some time.
Thus polynomials are studied like this and rational functions and what else you can do to extend the set of polynomials is studied separately.
