nitpicking the definition of a polynomial function A textbook I'm reading says that $f(x)=0$ is NOT a polynomial function, yet $g(x)=8$ IS a polynomial function since $g(x)=8=8x^0$ which satisfies the non-negative integer degree requirement.  Yet, it's still a monomial!  Or, can it be considered $g(x)=8=8x^0=8x^0+0$ which is now technically a POLYnomial ?
Also, $f(x)=0$ is not a poly by the same logic since it CAN be written as $f(x)=0=0x^{-3}$ which doesn't fit the poly definition above.  (Counterexample)
Edit:  The book says the zero function is not assigned a degree, while the nonzero constant function has a degree of 0.
 A: OK. But $f(x) = 0 = 0x^{-2} + 0x^{-1} + 0x^0 + 0x + 0x^2$ just as well as $g(x) = 8 = 0x^{-2} + 0x^{-1} + 8x^0 + 0x + 0x^2$. So this definition of polynomial seems difficult to justify.
The part about having no non-zero coefficients to negative powers of x is important because it creates discontinuity at x = 0. But $f(x)$, as defined, has no such discontinuity.
As you point out, monomials are generally considered a subset of polynomials. Rather than poly meaning many, as in two or more, polynomial is generally considered to be "some number of terms" and one is an OK number. This brings me to the one justification I can think of for excluding $f(x) = 0$. Because all coefficients are 0, this function really has 0 terms (It's the unnomial!) and you could say that zero is not quite enough terms to join the polynomial club.
Re-edit: It has come to light that the original question is a misread of the text book. The zero function is considered by the author (as by the rest of the world) to be a polynomial, but one with no degree, or an undefined degree. This makes much better sense. The unnomial is a member of the polynomial club.
A: The problem with your book's choice of definition is that, by that definition, the sum of two polynomial functions may not be a polynomial function. This is, generally speaking, a property one would really like to hold. I'm not going to go so far as to say the book is wrong (any author can define things however they like), but it seems a very unfortunate choice.
Edit: Most agree that the zero function is a polynomial function. The matter of degree is another story. It may be considered to have no degree or to have a degree of $-\infty$. The latter choice makes some theorems easier to state.
A: The book says the zero function is not assigned a degree, while the nonzero constant function has a degree of 0.
