Rhonda T.

asked • 06/12/14

Why is G(x)= 0 not a polynomial?

Working with what is and is not a polynomial function.

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Jose S. answered • 06/12/14

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Hello Rhonda,
I have to say initially I was going to say G(x) = 0 is definitely a polynomial! 
Then I looked up the definition, just to be safe and the most widely used definition seems to be that a polynomial must have more than 1 term.  I don't really understand why this definition is preferred(except for the fact that the root of the word poly is "many").  I have seen plenty of cases where constants are treated as polynomials of degree 0(the degree of a polynomial is the highest exponent the variable is raised to.   Hence 0 = 0x^0 = 0*1 = 0.  Or for any constant c, cx^0 = c*1 = c.

I'm sorry to provide such an ambivalent answer, but I hope I gave you at least one good reason to use for saying G(x) =0 is not a polynomial.  

Id just point out that in mathematics definitions are very important so whatever class you are taking, figure out the definition of a polynomial and see if G(x) = 0 meets that definition.  
 
Hope this was helpful,
Jose
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Jose S.

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Just another quick comment, Wikipedia says that a function is a polynomial function if it satisfies:
f(x) = a_nx^n +a_{n-1}x^{n-1) +... +a_1x + a_0
note that the last two values are equivalent to a_1x^1 and a_0x^0 = a_0*1 = a_0
The only restrictions are that n is not negative and that the a_i are constant coefficients. By this definition G(x) = 0 is a polynomial function, but again check the definition you are using in class.   
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06/12/14

Bob A.

I'd say because if it has only one term, it is a monomial.
A polynomial of degree zero (one term) in a way doesn't make sense.
 
It is just in the way mathematicians have defined the word.
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06/12/14

Jose S.

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I'd say that it makes sense precisely because of why mathematicians defined it that way.  It wasn't just a whim but because it is mathematically the most convenient definition.  Similar to how multiplication of complex numbers is defined in a way that makes things work nicely. 
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06/12/14

Darlene N. answered • 06/12/14

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Anything with "poly-" as a prefix always means more than one, whether in math or any other use (polygon = more than one side; polygamy = more than one wife; etc). The root word "nomial" means name or term. So just looking at the word, polynomial means more than one term.
 
Any function that is constant or consists of only one term is called a monomial, since "mono-" means one. I've always liked when mathematicians stick with words that make sense in regular language. :)
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Jose S.

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I think you are mistaken in your view of how mathematicians define polynomial and polygon.  While the definitions you give more or less capture the spirit of the mathematical definitions, for mathematicians it must be more precise.  A polygon has to be a plane figure for instance, which you did not include(taking things as assumed is not allowed in a mathematical definition).   An angle is a plane figure with more than one side but it is not a polygon because it is not closed.  Again, you may have thought it would be assumed but that is because you do have an understanding of what a polygon is and weren't trying to give a mathematical definition.  
"Poly," according to dictionary.com comes from the greek for "many" not simply "more than one. I don't  point this out to correct you really so much as to show why definitions based just on the word origin are not precise enough to be considered mathematical definitions.   How many is "many?"  Perhaps binomials should not be considered polynomials...two terms isn't "many." 
 
I stand by my advice to Rhonda that she consult the definition given in her textbook.  It is possible that the book agrees with you and says that a polynomial must have more than one term.  I would appreciate it if she got back to us with which definition her book uses since I am pretty curious about it.
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06/12/14

Darlene N.

Yes, you are correct in that mathematicians are very precise in their definitions, and I never discounted that. However, every word in mathematics is used because of the application of its origin. Yes, mathematicians define a polygon as a plane closed figure with more than one linear side. But they chose the word "polygon" because it means "many sides", which fits the figure albeit mathematically incomplete. Mathematical definitions are extremely precise because they are scrutinized mathematically (which means thoroughly, empirically, logically, quantitatively, etc etc etc).
 
My point in saying what I did wasn't dismissing any precise mathematical definition. It was just making the idea of a polynomial more accessible. Textbooks aren't perfect and can disagree, but the idea of a polygon having more than one side applies regardless. Technically, a polygon has to be composed of 3 or more line segments in the same plane with each segment intersecting with exactly 2 others at its endpoints and no segments intersecting anywhere but the endpoints. But to say it's a figure with many sides throws out a lot of figures, just like saying a polynomial has more than one term throws G(x) = 0 out of the definition of "polynomial."
 
I could say that a polynomial is an expression with three or more finite terms composed of a combination of constants, variables, and exponents such that there is no division by a variable and only whole number
exponents, which is its precise mathematical definition. But why say all that to answer this specific question when we can simply take the gist of the word "polynomial" and say it has more than one term? Plus in learning that, one relates the mathematical word to other English words, which means we've made a connection outside of mathematics -- always good. I save the precision for higher mathematics, where those details are necessary. The fact that a polynomial has more than one term is all that is needed to answer this question, and that fact doesn't change as we add the mathematical conditions to the technical definition.
 
Poly, by the way, does translate directly from the Greek as "many." However, when used in the form of a combining prefix (as in combining faces or segments or terms), the Indo-European translation is "more than one" (see freedictionary.com as one example, but this is a common idea).
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06/12/14

Jose S.

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I agree that it is always good to connect ideas to things outside mathematics(in my answer I also included that poly comes from many).  

The reason I included the facts definition for a polynomial that I did I don't think high school students should be intimidated by math.  I'm not saying you disagree, just explaining my own reasoning.  I find that explaining concepts as clearly as possible helps students better learn and understand them, rather than fear them.  Trying to explain the concept of a polynomial seems a worthwhile goal even if the question was only about G(x) = 0. 
 
Thanks for responding to the comments, I honestly appreciate the feedback and seeing how others think. 

I have a question though, you say you could have defined a polynomial as: " an expression with three or more finite terms composed of a combination of constants, variables, and exponents such that there is no division by a variable and only whole number exponents." 
May I ask where you got this definition?  Did you write it out yourself?  I'm only curious because as insinuated in my original answer, for years I have been used to the definition in the wikipedia article I linked to(and looking through some of my old textbooks it is explicitly said that all coefficients may be 0).   I understand of course that context matters, but that is precisely why I suggested Rhonda look at the definition in her own textbook.  I have no doubt different textbooks do provide different definitions, though I doubt any one is any more "precise" than others.   After all definitions are there so we know exactly what we're working with.  Going back to my question I wonder in what context that definition was found.  
 
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06/12/14

Darlene N.

I don't think high school students should be intimidated by it, either. As a long time high school teacher, curriculum writer and current doctorate student, I've found that it's easier for students to grasp a mathematical idea by building it slowly. For example, if Rhonda is in Algebra 1 or Algebra 2, she likely will not find a technically mathematical definition in her textbook because there's no need at that level to scrutinize it completely. Saying it's an expression with several terms involving x and some whole number powers is usually enough, as that's all she'll work with. Moreover, for this particular question, saying it has several terms deals with the issue. Not until Pre-Calc is the technical mathematical definition given or needed.
 
As for that definition, it was what we refined in both my number theory and modern algebra classes in college years ago. Luckily my prof was a big education guy (in addition to his straight math degrees) so his goal was not only to be technically correct but also clearly understandable in layman's language. We worked hard as mathematicians to verify that definition in the context in which the word polynomial is used. However, you can find a similar definition on www.mathisfun.com, or in several textbooks. I just found this in my Mathematical Ideas textbook: "A polynomial is a finite sum or difference of terms with only nonnegative integer exponents (aka whole numbers) permitted on the variables."
 
As for whether a binomial qualifies as a polynomial...that's kind of the same debate as whether a parallelogram is a trapezoid. It depends on whether a trapezoid is defined as "at least" one pair of parallel sides or "exactly" one pair of parallel sides. If it is the former, a parallelogram is technically a trapezoid. In my studies, I've always leaned toward using the most specific or clear term, meaning that I'd use the latter definition for a trapezoid -- exactly one pair. That way there is no confusion on what a true trapezoid looks like. The same is true here. Should a polynomial be defined as two or more terms or three or more terms? In truth, it doesn't change any operation you do with the polynomial/binomial. So I've always used the term "binomial" to mean two terms, distinguishing it from a larger polynomial. That way when I talk about factoring a binomial out of polynomial, my students know exactly what I mean. However, one could easily define a polynomial as "two or more" rather than "three or more" like we did.
 
So...short answer...that definition is found in several textbooks as well as through my own mathematical explorations under the guide of a professor of mathematics.
 
I've found several mathematical mistakes in Wikipedia, so I don't rely on it for technical definitions. If I need a general word definition, Wikipedia might do the job (as well as including someone's version of the mathematical definition). Unfortunately, as much as we want to think math is black and white, mathematicians still find room for debate. Guess it's in our nature, since we're so used to proving stuff. :)
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06/13/14

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