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/
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