How are monomials also polynomials?

Question

I was taking an online test and the question was "Monomials are polynomials? True or False?" I put false and it said I was incorrect. How are monomials polynomials?

Answer 1

It's misleading to say, " x^4 (a monomial) is secretly x^2 * x^2 (a polynomial)". x^2 * x^2 is not a binomial or trinomial or any other variety of polynomial besides a monomial; breaking it into a product does essentially nothing.



As a PhD student in math, I have never heard any debate whatsoever among actual math people about the meaning of the term "polynomial". The dictionary.com definition referenced in another answer which requires two or more terms is not remotely standard. dictionary.com actually gives several contradictory definitions; the referenced one is just plain wrong and should be updated.



Polynomials are finite sums of "terms" of the form ax^k where a is a constant, x is a variable, and k is a non-negative integer. The number of terms is 0 or more; 0 terms gives the 0 polynomial; 1 term gives a monomial; 2 terms gives a binomial; etc., though all of these are polynomials.



It's common to hear the phrase "non-constant polynomial" or "non-zero polynomial" in advanced math, so both monomials and the 0 polynomial are typically thought of as "polynomials". It would be extremely inconvenient if monomials weren't thought of as polynomials since they share all the usual properties and theorem statements would constantly have to say "polynomial or monomial", which is silly.

Answer 2

Definitions appear to vary. At dictionary.com, the first definition that comes up specifies "two or more terms", as do several others, but the definition from the World English Dictionary does not specify multiple terms. However, the Wikipedia entry for polynomial explicitly indicates that a monomial is a class of polynomial.



Based on the meaning of the prefix poly-, it would seem that a monomial should not be considered a type of polynomial, any more than monogamy is a type of polygamy. You could certainly make a good argument that your answer should have been considered correct.



*Edit*: A polynomial definitely does not need to have four or more terms. Two terms is sufficient for there to be no question that the "poly-" prefix applies (and to satisfy the several dictionary definitions that indicate a polynomial has two or more terms.).

Answer 3

Well it's math, not English. Polynomials encompass monomials, binomials, and trinomials etc according the algebraic nomenclature, and if you think about it, even a monomial such as x^4 (a monomial) is secretly x^2 * x^2 (a polynomial).

Answer 4

In mathematics, a monomial is roughly speaking, a polynomial which has only one term





Examples,



x^2 is a polynomial, but can be further described as an "mono"mial.



x^3 + 5 is a polynomial, but can be further described as an "bi"nomial.



x^4 +3x+4 is a polynomial, but can be further described as an "tri"nomial.



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Defn of a polynomial:http://en.wikipedia.org/wiki/Polynomial





Defn of a monomial:http://en.wikipedia.org/wiki/Monomial