Definition of MonomialDate: 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 polynomials. 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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