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### Definitions of Monomials and Polynomials

```Date: 03/11/2004 at 13:29:56
From: Kathy
Subject: Monomials

Why is 1/x considered not to be a monomial?  It could be written as
x^(-1) which is just one term.  My math text says that it is NOT a
monomial but it does not say why.

```

```
Date: 03/11/2004 at 14:19:52
From: Doctor Peterson
Subject: Re: Monomials

Hi, Kathy.

Something is a monomial if it fits the definition.  How does your text
define the word?  (Not all texts give good definitions, or follow them
properly!)

A monomial is a term in a polynomial, or equivalently, a polynomial of
one term.  A term, in this sense, is a product of constants and
variables that includes positive powers, but not negative powers
(which represent division by a variable).

Here is one definition of "monomial":

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

A polynomial consisting of a product of powers of variables
together with a coefficient in some unit ring R.

Now, if you are asking why we define a monomial this way, excluding
negative powers of variables, there may be more to discuss; but it's
basically because defining polynomials that way makes for nice math;
polynomials as defined have useful properties that would be lost if
we changed the definition.  When we start dividing by variables
(rational functions), we find we can always put the expression in the
form of a ratio of polynomials (with positive powers); if polynomials
included negative powers it would be harder to describe the standard
form of either.

Why do we make this restriction?  That's just the definition, and
it can be hard to say exactly why a definition is what it is.  But
I would guess that it arises from (a) the historical origin of
polynomials, and (b) the usefulness of such a definition in
various contexts.  I'm not prepared to give a full defense of the
definition, and don't even find any mention of its origin in the
few math history books I have at my disposal.  But my suspicion is
that (a) it originated before negative exponents were considered,
so that exponents were thought of as merely a shorthand for
multiplication, and a monomial can be considered as involving
multiplication ONLY; and (b) it is useful in contexts in abstract
algebra where division and negative exponents need not even be
defined!  Moreover, many theorems involving polynomials would not
apply to the broader definition.  This is often what motivates a
definition: it ties together a set of objects that belong in the
same theorems.

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 Definitions
High School Polynomials

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