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Definition of Monomial

Date: 04/26/2004 at 14:31:46
From: Marcy
Subject: Monomials

Do monomials include quotients of variables?  For example, would x/y 
or xy^-2 be considered a monomial?

I have always thought that monomials included quotients of variables.
However, the text book I am teaching from this year says that x/y is
not a monomial, because monomials only include the products of
variables or variables and constants.  Any division of variables or
variables with negative exponents would not be a monomial.  Which
definition is correct?

Also, if x/y is not a monomial, does that mean that x/y + 3 is not a 
binomial?  Thanks!

Date: 04/26/2004 at 14:59:22
From: Doctor Peterson
Subject: Re: Monomials

Hi, Marcy.

A polynomial is a sum of terms, each of which is a PRODUCT of 
constants and variables, and therefore can be written as a coefficient 
(possibly 1) times a product of (positive) powers of variables 
(possibly none).  So xy is a monomial, but x/y is not; and likewise 
x/y + 3 is not a binomial.

The main reason for this, I think (apart from the historical fact that 
when polynomials were first studied, negative exponents were somewhat 
disreputable, if not entirely unknown) is that many important theorems 
depend on having only positive exponents.  If we allowed negative 
exponents, polynomials would not be continuous and defined everywhere, 
which is an important property.  Other theorems about polynomials, 
such as those concerning the number of zeros, would similarly fail 
under the broader definition.  Also, this definition works in many 
situations where the variables can take on values that are not 
numbers, and which can't be raised to negative powers, making it 
meaningful in abstract algebra.

But it is easy to forget this and accidentally extend the definition 
to allow negative exponents, since we often just talk about terms as 
products of powers (without specifying "positive" powers); and some 
books or teachers probably do that.  In fact, it's been done within 
the Dr. Math archives!

Functions like polynomials, but with negative powers, are called 
rational functions, and can always be written as the quotient of two 

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

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Definitions
High School Polynomials

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