Associated Topics || Dr. Math Home || Search Dr. Math

### 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/
```
Associated Topics:
High School Polynomials

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search