Is (x^(1/2))^4 a polynomial?

Date: 06/10/2002 at 22:17:44
From: Mario Ragasa
Subject: Is (x^(1/2))^4 a polynomial?

Is (x^(1/2))^4 a polynomial or not?  The head of my math department
said it is not since it is under the square root sign, but I said we
have to simplify first to make it a polynomial. What is really the
convention for deciding whether something is a polynomial? Thanks!

Date: 06/11/2002 at 08:48:01
From: Doctor Peterson
Subject: Re: Is (x^(1/2))^4 a polynomial?

Hi, Mario.

You can't really simplify this expression to a polynomial; in doing 
so, you lose the fact that it is not defined for negative x. (That is 
true if you are working with real numbers, because you can't take the 
square root; and also if you allow complex numbers, because then 
"the" square root is not defined; there are two roots, and no way to 
define a single principal root.) So your function does not have the 
right domain to be a polynomial. The same would be true for

    x^2 - 1
    -------
     x - 1

Strictly speaking, a polynomial is a specific _form_ of expression, 
not just any function equal to such an expression; I wouldn't even 
say in this strict sense that x(x+1) is a polynomial, because it is 
not written in polynomial form. In cases like this the work is so 
trivial that we usually don't bother to distinguish between "being a 
polynomial" and "being able to be written as a polynomial", but there 
is still a difference. See the definition here, which calls a 
polynomial an expression of a certain form, not a function with a 
certain behavior:

      http://mathworld.wolfram.com/Polynomial.html 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/