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A rational function is the quotient of two polynomial functions.
A power function is of the form xa where a is a real number. A polynomial function is of the form anxn + an-1xn-1 + ... + a1x + a0 for some positive integer n, and all the ai are real constants.
As with most advanced math, if your "real life" involves engineering work, you will use such math; otherwise, you will hardly have anything to do, in this case, with polynomial functions.
Basically, an expression is not a polynomial when anything is done that is not allowed in a polynomial - for example, use any variable in the denominator of a monomial, use non-integral powers or radicals (which is basically the same as a non-integral power), use functions, etc.