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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.

Here is what I wrote to someone else who asked about this:

  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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