For integer x only, is x considered a polynomial?

[tgt]

For integer x only, is x! considered a polynomial?

 

[St. Aegis]

no 4char

 

[Diffy]

well,
x! = x * (x - 1) * (x - 2) * ... * 2 * 1

And for any integer x! = n where n is some integer.

So, is an integer a polynomial?

 

[HallsofIvy]

well,
x! = x * (x - 1) * (x - 2) * ... * 2 * 1

And for any integer x! = n where n is some integer.

So, is an integer a polynomial?

That's a strange response! For any integer, x, x²= n where n is some integer.
Is an integer a polynomial? No, of course not. But that has nothing at all to do with whether or not x² is a polynomial, which it is.

x!, for x a positive integer is not a polynomial but for a very different reason: because it does not satisfy the definition of "polynomial".

 

[Diffy]

From Wikipedia:
A function ƒ of one argument is called a polynomial function if it satisfies

for all arguments x, where n is a nonnegative integer and a0, a1,a2, ..., an are constant coefficients.

I think integer fits this definition since n is allowed to be 0, no?

 

[HallsofIvy]

Yes, you can think of the constant function, whether or not it is an integer, as a polynomial. But that still has nothing to do with the question. Surely you are not confusing a function with a value of the function?

 

[Diffy]

I would think that is exactly what I am doing. I apologize tgt if I mislead you.