Talk:Polynomial

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[edit] Definition

Shouldn't the intro say "non-negative whole number exponents" rather than positive? —Preceding unsigned comment added by 65.82.119.254 (talk) 17:16, 27 November 2007 (UTC)

Not necessarily. The into does not describe formal or reduced polynomials, so it allows constant terms simply by leaving it optional whether to exponentiate at all. –Henning Makholm 22:31, 27 November 2007 (UTC)


The articles definition is currently inconsistent with respect to the zero-polynomial, p(x) = 0. The beginning of the overview states 'Polynomials are built as a sum of zero or more terms called monomials', while a few lines further down: 'A polynomial is a sum of one or more monomial terms'. Assuming that the zero-polynomial is indeed included in the definition, I think the 1st paragraph should read: 'A polynomial is a sum of zero or more terms called monomials' (i.e. singular and no use of 'built'). Lklundin (talk) 20:52, 13 January 2008 (UTC)
I had applied such a correction and several other changes, also introducing the term "zero polynomial",[1] but this got all reverted. Since I believe that most of my changes were real improvements, I'm now re-applying my changes one by one, which, unfortunately, may introduce temporary inconsistencies.  --Lambiam 01:31, 14 January 2008 (UTC)

While there is a technical sense in which the zero polynomial has no terms at all (zero terms) this is bound to confuse the beginner. As a compromise, I suggest that we restore the older and easier to understand intro, which avoids the issue, and discuss this technicality in the more technical sections of the article. Rick Norwood (talk) 16:04, 14 January 2008 (UTC)

Is x + 1 a polynomial? As the article stands now, the reader should conclude that it is not. What is the degree of the polynomial 0x3 + x? As the article stands now, the reader should conclude that it is 3. And of which monomial(s) is the zero polynomial the sum? I am all in favour of simple presentations, but not if this goes at the expense of correctness. Otherwise we might as well simply define a polynomial as "a bunch of numbers and x's".  --Lambiam 17:55, 14 January 2008 (UTC)
I agree with Lambian. The correct definition of a polynomial expression, the correct treatment of the zero poylnomial, and a proper distinction between expressions and functions are not difficult to understand. In this case it will assist the beginner if we give them the correct definition from the outset, rather than starting with a vague and inconsistent definition which has in any case to be corrected later. Gandalf61 (talk) 09:39, 15 January 2008 (UTC)

I strongly agree that a simple definition must also be correct. It must be the truth -- just not necessarily the whole truth. I also agree that the distinction between an expression, an equation, and a function needs to be made not only here, but also consistently throughout Wikipedia. The only point of disagreement is on whether 0 has zero terms or one term. The references I've looked at all talk about the number of non-zero terms, rather than the number of terms. Rick Norwood (talk) 14:29, 15 January 2008 (UTC)

I reread the beginning of the article with Lambiam's examples in mind, and see his point about 0x^3 but not his point about x + 1. I'll give it some more thought. Rick Norwood (talk) 14:33, 15 January 2008 (UTC)

    • DO YOU THINK ALL OF YOU COULD CONTINUE YOUR REDICULOUS BANTER IN YOUR PERSONAL EMAILS. AS A STUDENT, I REALLY DIDN'T FIND ANY OF THIS ARTICLE SUFFICIENTLY HELPFUL BECAUSE YOUR EGOS WERE TOO DISTRACTING!)) —Preceding unsigned comment added by 12.213.112.58 (talk) 21:33, 13 April 2008 (UTC)
  • Please do not SHOUT or make personal attacks per WP:TALK. If you don't understand, use the Simple English version. Regards, Lunakeet 12:27, 6 May 2008 (UTC)

[edit] Edits in early March 2008

I made an edit today whose main purpose was to replace the frequently used notion "monomial" by the more unambiguous notion "term", but in the mean time made some other changes that I thought were improvements. This provoked a message by Gandalf61 on my talk page from which I conclude I may have hit something more sensitive than I expected (see the discussion there). I will probably soon improve some formulations. There is I think a fundamental difficulty, in that one cannot be both light in formulation and 100% correct. Formally it is hard to avoid making a destinction between polynomials and expressions describing polynomials: when writing (X+1)2=X2+2X+1 I'm writing distinct expressions to designate the same polynomial, which is the whole point of the exercise (and I am not affirming equality for all values of X). But this destinction should not be pushed to make the discussion opaque for the non initiated. The same goes for zero terms, I should be able to say that a polynomial doesn't change if I drop zero terms (which I cannot even start to say if a term is not allowed to be zero) but obviously something changes when I do so. My current idea is that the introduction should start by saying what a polynomial naively is (a sum of certain kinds of terms) then say that some manipulations are common to rearrange polynomials (gathering similar terms for one thing), that other common operations allow expressions like (X+1)2 that seem to contradict the earlier definition, and finally some question could be raised (but maybe should better not) of what the value of a polynomial (the thingy that remains unchanged under our manipulations) should be defined to be. Marc van Leeuwen (talk) 20:54, 7 March 2008 (UTC)

Well that is done. Or rather was until User:24.96.130.30 (talk) came back to work. Does anybody know how to integrate the precious contributions from this user? Marc van Leeuwen (talk) 18:24, 8 March 2008 (UTC)

(reply to, now erased, message on User:24.96.130.30 (talk), copied from Marc van Leeuwen's talk page):

Marc, I redid the first two parts of the article because they were very unorganized, and the definition of a polynomial was very poor. It was not my intent to permanently leave out your edits, but I was a little upset that you did a complete undo and I did not have the time to re-include your edits. I will try to go back and re-include your edits. My apologizes. 24.96.130.30 —Preceding unsigned comment added by 24.96.130.30 (talk) 20:44, 8 March 2008 (UTC)
You say the first two parts of the article were very unorganized, and the definition of a polynomial was very poor. I do not contest improvements were possible (which is what I tried to do), but I fail to see the essence of your point, particularly in comparison with what you wrote. The definition used to say "a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents." Apart from a somewhat unfortunate wording concerning exponents (I would prefer "raising to positive integer powers"), this is exactly the way polynomials can be built up. You replace that by "a polynomial is an expression involving a sum (or difference) of a finite number of terms. Each term is comprised of one or more variables raised to a non-negative integer and multiplied by a coefficient." While more lengthy, this is certainly not more precise, in fact it is widely too restrictive and at some points unclear. As a very first description, vagueness is less harmful than being too restrictive, as precisions can be added in later sections, but a too narrow idea is hard to widen later. As for your restrictiveness, here are some examples of non-polynomials according to your description: 4X7 (doesn't involve a sum at all), (X + 1)2 (involves a sum but is not one), 4x5 + 7x (the x in the second second term is not raised to any power), − 7x6 + x2 (the second term has no coefficient), 3x4 + 7x3 − 6 (the final term lacks variables). These may seem pedantic criticisms, but keep in mind that the introduction is there to explain the notion to people possibly unfamiliar with it, who might not know for instance that a term without coefficient is considered to be the same as the term with coefficient one. As for unclarity, you do not for instance say that the powers of variables "comprised" in a term are supposed to be multiplied together. On the other hand "finite" might be a useful addition for those who are used to series (even if they probably already know what a polynomials is), although I think that the "expressiion constructed from…" for the old definition is implicitly finite (as is anything we can completely write down).
Concerning organization, I doubt whether your changes are real improvements. There is now, instead of "Overview" a section "General Form" which gives only a specific form for polynomials in one variable, calling them P(x) for no apparent reason (leaving that out twice would result in a clearer statement), and which suggests that such polynomials necessarily involve a \sum sign, and for those who understand that notation, that the terms must involve a continuous range of powers of x starting at 0. It then goes on about power series and Laurent series. Saying "polynomials are closely related to power series and Laurent series" is like saying "rational numbers are closely related to real numbers and complex numbers"; nobody would contest the truth of that, but it is hardly a helpful point of view when trying to understand polynomials (or rational numbers).
The section "Canonical form" does mention the possibility of multiple variables, and says there canonical form is obtained by listing terms from highest degree to lowest degree. While that is possible, it does not have a unique outcome as the word "canonical" would suggest. To really nail down a unique canonical form for general polynomials is possible but requires more technical effort, which is probably not a useful exercise so early in the article. (In the section "Form of a polynomial" I did a modest but honest (and therefore somewhat lengthy) effort to describe all steps needed in general to get to a "standardized form", and this form is still not canonical as it ignores questions of reordering terms and factors within a term.)
Finally, I found your section "Classifaction of polynomials" amusing, but not really appropriate so early in the article. It is nice to know what a hectic polynomial is, but not essential to understanding polynomials in general; remember that this article is about mathematics, not taxonomy.
Marc van Leeuwen (talk) 09:52, 9 March 2008 (UTC)

I agree with Marc that the rewrite does more harm than good. My major objection is that it is unclear to the beginner, with all those zero exponents, and a lot of waffling. I think we need to go back to the earlier version, and if it needs modification, discuss changes here before doing a masive rewrite of a stable article. I'll wait 24 hours, to see if there are any objections to a return to the status quo ante, followed by more gradual changes. Rick Norwood (talk) 14:19, 9 March 2008 (UTC)

That's OK with me. I prefer that somebody else than me undoes those changes, as I don't want to get in an undo-fight with User:24.96.130.30. But please consider whether to revert to before 7 March, or to the version (with oldid=196801822, I don't know how to make a link to that Marc van Leeuwen (talk) 06:14, 10 March 2008 (UTC)) at the point where my edits of 8 March (including a subdivision of the "Overview" section) were undone by the mentioned anonymous user, hijacked by his effort to improve the article in a different way. I really did spent quite some effort to give a neutral introduction, understandable for beginners and respective of different sensibilities, yet formulated without sloppiness (though I'll plead guilty to some lengthyness). So I am rather disappointed that all this disappeared in an unfortunate way before anybody else could take a look at it. Marc van Leeuwen (talk) 22:06, 9 March 2008 (UTC)
To make this a little more inclusive, would someone involved mind providing links to the various preferred versions? Silly rabbit (talk) 22:15, 9 March 2008 (UTC)


Marc, I'd appreciate it if you would leave the sarcastic remarks out of this discussion. Phrases like "precious contributions" and "Classifaction of polynomials" amusing..." are not appreciated. Let's keep this discussion on a professional level. Otherwise, I appreciate your feedback. Now, let me try to address the above comments:

I found two major problems with the original text that prompted me to rewrite the first two sections. The first was the definition of a polynomial, and the second was the lack of organization to the Overview.

First, there were two different definitions for a polynomial, one in the intro and the other in the Overview. The intro definition was poor, specifically the first sentence. For example, it lacked the phrase "finite number of terms (or monomials)" which distinguishes it from other series as I tried to show later when comparing it to the Laurent series and power series. The definition in the overview was better, but way too verbose for an intro. I don't believe we need two different definitions in two different places.

>"I would prefer "raising to positive integer powers")."

I orginally was going to use the phrase "positive integer", but elected to use "non-negative integer". I wanted to make sure readers knew that it included zero, which is used in subsequent examples.

My point was not about positive versus non-negative, it was about using the verbal form "raising to the power" that blends better with "addition" and "multiplication" than the noun "exponents". A very minor style criticism. Marc van Leeuwen (talk) 07:01, 10 March 2008 (UTC)

>"As a very first description, vagueness is less harmful than being too restrictive, as precisions can be added in later sections."

Point taken. But I think we need to be careful with the vagueness in a definition to the point the reader confuses polynomial with some other series.

>"As for your restrictiveness, here are some examples of non-polynomials according to your description: 4X7 (doesn't involve a sum at all), (X + 1)2 (involves a sum but is not one), 4x5 + 7x (the x in the second second term is not raised to any power), − 7x6 + x2 (the second term has no coefficient), 3x4 + 7x3 − 6 (the final term lacks variables). These may seem pedantic criticisms, but keep in mind that the introduction is there to explain the notion to people possibly unfamiliar with it, who might not know for instance that a term without coefficient is considered to be the same as the term with coefficient one."

The equivalent expressions address this concern. I think pedantic criticisms may be found with any definition. For example, I don't believe all people reading the phrase "using only the operations of addition, subtraction, multiplication" will know that division may apply e.g., x2 / 7. It wasn't clear to me why it included the inverse of addition, but not the inverse of multiplication.

You are right that subtractions are not absolutely necessary (although everybody uses them all the time in polynomials). But division does not have a similar status to subtraction in polynomial expressions. One could add the phrase "division by nonzero constants" to the list however. This is of minor importance.
More fundamentally I think the difference of point of view is whether one wishes to describe a very strict and formal standard form as the first description, and than expand later by equivalence, or rather start with a comprehensive description of polynomial expressions and later show how they can be reduced to a standardized form. You prefer the former approach, I prefer the latter. For me showing how polynomials can be standardized (not an entirely trivial subject) should be an important part of the theory of polynomials, but not of their initial description in an encyclopedia. Marc van Leeuwen (talk) 07:01, 10 March 2008 (UTC)

>"As for unclarity, you do not for instance say that the powers of variables "comprised" in a term are supposed to be multiplied together."

I don't, and wouldn't, because they aren't. Powers are added. I guess I'm missing your point.

You are mistaken. In a term comprised of the powers x3 and y2 they are multiplied together, not added.

>"On the other hand "finite" might be a useful addition for those who are used to series (even if they probably already know what a polynomials is), although I think that the "expression constructed from…" for the old definition is implicitly finite (as is anything we can completely write down)."

A definition should be clear to the reader, not just those that already know what a polynomial is. I certainly don't believe "finite" is implicit. If we are going to say it's implicit, I could argue that some of the earlier comments are implicit too such as: "...who might not know for instance that a term without coefficient is considered to be the same as the term with coefficient one" and "4x5 + 7x (the x in the second second term is not raised to any power), − 7x6 + x2 (the second term has no coefficient), 3x4 + 7x3 − 6 (the final term lacks variables)" You can't have it both ways.

I think being finite is implicit in many notions, and in particular in the notion of "expression". Try writing an infinite expresion. The summation operator used to describe infinite summations is not just a use of addition, it requires special considerations to introduce (and more formally, a set closed under addition is not required to be closed under infinite summations). I honestly don't think anybody would think the original decription was meant to include \sum_{i=0}^\infty x^i. Note by the way that while you mention that the number of terms must be finite, you do not mention that the number of powers of variables in each term should be finite as well, which show you did not automatically consider the possibility of infinite products like you did for infinite sums. Marc van Leeuwen (talk) 07:01, 10 March 2008 (UTC)

>"calling them P(x) for no apparent reason (leaving that out twice would result in a clearer statement)"

I agree. So why didn't you make the change and delete the P(x)? The current version shows an expanded form which I actually prefer over the sigma notation. In any case, the previous version lacked this.

>"It then goes on about power series and Laurent series. Saying "polynomials are closely related to power series and Laurent series" is like saying "rational numbers are closely related to real numbers and complex numbers"; nobody would contest the truth of that, but it is hardly a helpful point of view when trying to understand polynomials (or rational numbers)."

I believe you are being too critical here and are making a judgement for all readers that is not true. Although I knew there were differences, I for one wasn't aware of the specific differences until I wrote the revision. Since the statement is true, as you agree, why remove something that readers may find helpful? Keep in mind many readers are not mathematicians or hold degrees, so additional info can be quite helpful in their understanding.

I cannot believe you did not learn about polynomials before you ever heard about power series and Laurent series. Maybe you studied functions being given by a convergent power of Laurent series before ever considering polynomial functions. But all of those presuppose a certain experience with manipulating finite expressions involving constants and variables, in other words (if division is not involved) polynomial expressions. The confusion between polynomials and polynomial functions is widespread (because of the way math education is organised), but should not IMHO affect this article. Marc van Leeuwen (talk) 08:56, 10 March 2008 (UTC)

>"While that is possible, it does not have a unique outcome as the word "canonical" would suggest. To really nail down a unique canonical form for general polynomials is possible but requires more technical effort, which is probably not a useful exercise so early in the article. (In the section "Form of a polynomial" I did a modest but honest (and therefore somewhat lengthy) effort to describe all steps needed in general to get to a "standardized form", and this form is still not canonical as it ignores questions of reordering terms and factors within a term.)"

Yes, it was too lengthy, and there were no examples. That was precisely why I rewrote and restructured it. I do not take issue with you on most of its general content. But its not in the reader's interest to have to read through a lengthy one paragraph discussion to try and find out what the canonical form of a polynomial is. (Canonical was not used, and 'standardized form' was not bolded.)

Excuse me, I had not recognised your section "Canonical Form and Degree" as a rewrite of my "Form of a polynomial" (which you undid). I avoided the (usually rather strict) notion "canonical" since it is hard to define one that is unique (ordering terms by weakly decreasing degree does not suffice), and people would probably not agree about its precise form, or even if the notion exists for polynomials at all. Please note that the "general form" of a polynomial in one variable mentioned earlier need not be a canonical form as you define it, as it may contain terms with zero coefficients, while I gather from your examples (rather than from their description) that canonical forms omit such terms. My "standardized form" is not universally accepted terminology either, whence I found it unnecessary to put in bold. If I had to define a canonical form, I would say: expression as a linear combination of monomials (power products). But that displaces the problem to describing the notion of a "linear combination" (certainly finite) of an infinite family of elements. Marc van Leeuwen (talk) 08:56, 10 March 2008 (UTC)

>Finally, I found your section "Classifaction of polynomials" ... not really appropriate so early in the article. It is nice to know what a hectic polynomial is, but not essential to understanding polynomials in general; remember that this article is about mathematics,..."

I disagree. Any discussion of polynomials will include their classification. An early introduction allows future writers to use the terms in subsequent sections without having to include a definition. Doing so would again require the reader to look through the entire article to find out what a hetic polynomial is. Classification is one of the first things students learn about polynomials. The table makes it easy for the reader to quickly see and compare.

I don't know about your students, but mine never learn about hectic polynomials. Apart from constant, linear, quadratic, cubic and very maybe quartic, there is little chance these notions come up in a discussion about polynomials. Note that these notions (even up to sextic) were already present (and in boldface) before you edited (as well as binomial and trinomial), so I cannot see what was wrong with that. Really, classification is one of the least important things one can do with polynomials. What do those huge tables really add to our understanding of polynomials? Marc van Leeuwen (talk) 08:56, 10 March 2008 (UTC)

>"I think we need to go back to the earlier version, and if it needs modification, discuss changes here before doing a massive rewrite of a stable article. I'll wait 24 hours, to see if there are any objections to a return to the status quo ante, followed by more gradual changes." Rick Norwood (talk) 14:19, 9 March 2008 (UTC)

I would prefer we wait and resolve the polynomial definition here before discussing any rollback, otherwise we may be back to where we are. I don't think we are that far apart. Once that is decided, it may resolve some of the other minor points. I still believe we need to include more examples, the two classification tables, and the general form (expanded series form--not the sigma). We can then decide whether we should include the revised definition and changes to the current version, or do a rollback and include the changes in it. That said, if there is a loud chorus of objections to the new version, I won't be a lone hold-out. —Preceding unsigned comment added by 24.96.130.30 (talk) 04:56, 10 March 2008 (UTC)

I think an important point issue is related to the way one pronounces the '=' in (X + 1)2 = X2 + 2X + 1.
I prefer to pronounce it "equals" while others prefer "is equivalent to". But I would also say that left and right hand sides are distinct polynomial expressions that designate the same polynomial. The relevance of this remark is that I think it is acceptable, and even preferable, to initially define polynomials as polynomial expressions, namely as (finite) expresssions formed from variables and constants by additions and multiplications (subtractions and raising to positive integer powers don't hurt either), as in the original description. But one can then very quickly add that all polynomials expressions can be worked out as a sum of terms. Since I belong to the "equals" pronouncers, I wrote, just after the description in the lede, that "A polynomial can be written as a sum of terms…", whereas the "is equivalent to" pronouncers would have to say "A polynomial is equivalent to a sum of terms…". But I insist that, whatever way one wishes to have it, (X + 1)2 is an honest polynomial, and I have serious problems with a description in the lede that excludes it. Marc van Leeuwen (talk) 09:23, 10 March 2008 (UTC)
To clarify my point, I rewrote the lede so that it is acceptable according to my convictions, and hopefully does not offend too many others. I think distinguishing polynomial expressions and polynomials right away, and adding that hardly anybody cares to keep up the distinction, may be necessary to prevent people getting into fights rewriting the definition according to their view. Marc van Leeuwen (talk) 14:07, 10 March 2008 (UTC)

[edit] Merge from polynomial sequence

The polynomial sequence article is currently a stub, with a list of various named polynomial sequences. I would like to move the textual content (it's only a few sentences) here to Polynomial. The list should then be merged to List of polynomial topics. This seems a more efficient arrangement of the information, since the polynomial sequences are, as far as I can tell, already listed at List of polynomial topics. Silly rabbit (talk) 14:11, 30 January 2008 (UTC)

The question is really is there much more to say about the topic? If it is just a set of polynomials indexed by integers then amerge is fine. However the classes of polynomials indicate that there is some scope for expansion, the concept of Orthogonal polynomials could benefit from a summary style section and I guess the other classes could to, further the article needs at least one example to illustrate the concept. --Salix alba (talk) 16:37, 30 January 2008 (UTC)
You raise a good point. As far as I can tell, Polynomial sequence really is just about polynomials indexed by the integers. It is the special cases of this, notably the orthogonal polynomials, which make the concept a valuable one. So I don't think there is much scope for expansion at the polynomial sequence article without developing into a content-fork of other, more important subjects. Silly rabbit (talk) 16:43, 30 January 2008 (UTC)
The Integer sequence article follows the same basic plan as Polynomial sequence, but we would probably not want to merge it into the Integer article, even though the latter is only about half the size of Polynomial. WP:MERGE suggests merging when a page is very short and is unlikely to be expanded, but what is "very short"? While the present Polynomial sequence cannot be expanded a whole lot, more could be said. On the whole, I'd say that this is marginal: there is no big gain or big loss in merging.  --Lambiam 05:09, 31 January 2008 (UTC)

[edit] Multivariate polynomials, Laurent series

Currently multivariate polynomials are only introduced (although this name is not mentioned) very near the end, in the section Extensions of the concept of a polynomial. Tucked away as this is, and expressed in a ring-theoretic way, one would be inclined to think that this is some concept of highly advanced maths, instead of the quite mundane concept it is, which is all over the place in many Wikipedia articles on elementary mathematical topics. The notion of Laurent series, on the other hand, gets prominent prime-time coverage. I feel this is the wrong way around.

A specific problem is that currently the notion of degree is defined for multivariate terms and polynomials, which have not been introduced at that spot. We should actually be clear at some point that the set of variables a polynomial is "in" is a parameter, and make sure the reader will understand that ax2 is linear in a, quadratic in x, and cubic in the variables {a, x}.  --Lambiam 14:28, 9 March 2008 (UTC)

The treatment of Laurent series here really bothers me. Personally, I would be happy if all discussion of Laurent series and power series be removed, or at least moved to a separate section at the very end.
I also agree that multivariate polynomials need to be given much better coverage. Someone should start a section Polynomials in several variables before the other sorts of generalizations are introduced. Examples of several variables should be presented in the new section, as well as the various notions of degree (homogeneous degree, degree in x versus that in y, etc.)
Finally, let's agree that the first part of the article should deal only with polynomials in one variable. Some parts of the article will need to be adjusted to reflect this. We can then deal at length with polynomials in several variables in a section of its own. Silly rabbit (talk) 14:41, 9 March 2008 (UTC)
I agree, but let's keep just this near the top:

"A polynomial in one variable has the general form:

c_N x^N + c_{N-1}x^{N-1} + \cdots + c_2 x^2 + c_1 x + c_0." —Preceding unsigned comment added by 24.96.130.30 (talk) 05:14, 10 March 2008 (UTC)

[edit] back to the future

This article is read often, especially by high school students who really want to understand polynomials. We need to cover the basics first, and save abstractions for later. I am going to revert to an earlier version, when the article was relatively stable, and then we can discuss changes as needed. As best I can tell, the only person who has gone on record as prefering the recent major rewrite is its author. Rick Norwood (talk) 14:34, 10 March 2008 (UTC)

Note that the version I've reverted to includes the following:

"An expression that can be converted to polynomial form through a sequence of applications of the commutative, associative, and distributive laws is usually considered to be a polynomial."

This, I think, takes care of the problem of "equals" vs. "is equivalent to" without getting needlessly technical.

Some of the changes made during the first week in March were probably improvements, but let's discuss them here, one at a time, before trying to rewrite the entire article. Rick Norwood (talk) 14:42, 10 March 2008 (UTC)

This version is better. The definition in the Overview is much less verbose. I made some minor edits that I believe are necessary. However, the definition in the intro needs some major work:

"In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. For example, x^2 - 4x + 7\, is a polynomial, but x^2 - 4/x + 7x^{3/2}\, is not a polynomial because it involves division by a variable and also because it has an exponent that is not a positive whole number."

Aside from the "positive whole number" language, the phrase, "using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents." is far too fague. The second sentence 'tries' to remedy this by illustrating what is, and is not, a polynomial. But this attempt fails to clarify the earlier phrase. Furthermore, the intro is not the place to show what a polynomial is not.

Certainly, we can do better than this for the lead-in sentence(s). Either I will take a stab at it, or someone else can. I don't care. It needs to be changed. —Preceding unsigned comment added by 24.96.130.30 (talk) 00:56, 11 March 2008 (UTC)

Please explain what you mean by vague (I suppose). I think it should be quite clear to anybody what those arithmetic operations mean. And so one can form expressions such as (x − 1)2(x2 + 2x + 2). That is a polynomial (even though it may not be in the form you think polynomials should have). If anybody disagrees with this statement, let us discuss that before starting to change the definition into something that excludes the given example (as did your previous rewrite). If not, please explain what is so bad about the current definition. For me it has the advantage that it is in completely elementary terms that do not require any previous knowledge about polynomials to be understood. As for my wishes, I would like the lede to include both a polynomial of the kind of the example I just gave, as well as its written out form x4 − x2 − 2x + 2; this would give some impression of the posisbilities of polynomials. By the way, I think examples should try to not be too simple, in order to avoid false impressions; but please say if anybody thinks my example would on the contrary be too complicated for readers. Marc van Leeuwen (talk) 10:02, 11 March 2008 (UTC)

I disagree that the intro is not a place for an example of something that is not a polynomial. In fact, I tell my students that each time the learn a new mathematical word or phrase, they need to be able to give not only an example to which the phrase applies, but also an example to which the phrase does not apply. When learning about groups, it is not enough to give examples of groups, you also need to give examples of mathematical objects that are not groups, and explain why. Rick Norwood (talk) 12:49, 11 March 2008 (UTC)

Concerning the definition in the intro... As I mentioned earlier, we have two definitions for polynomial----one in the intro and another in the Overview--I prefer we have only one. I think I understand your reason for this, but it creates several problems.

First, notice we say addition, subtraction, and multiplication, but leave out division. This immediately signals the reader that division is not allowed. We go on to reinforce this notion by giving an example that uses division and stating it is not a polynomial. If the definition left out subtraction, then it would be a slight improvement. However, the example showing what a polynomial is not is introduced too early. It should be introduced after the more rigorous definition is given in the Overview for support.

Now, if some young readers were asked after reading the intro whether divisions were allowed in the constructions of polynomials, I suspect that some, or many, would probably say no. When the reader continues reading on to the Overview, it too doesn’t mention division (and rightly so). This further reinforces the misconception that divisions are not allowed. But of course, we know that divisions are allowed in certain cases such as x2 / 7.

Second, by having two definitions we will have to always check to ensure they are in harmony. If one makes a change to one, someone will need to ensure the same change is made to the other (e.g., changing the word ‘’term’’ to ‘’monomial’’). Most math pages don’t have two definitions. I don’t believe we need two.

Third, even when the two definitions are in harmony, a reader ‘’could’’ mistakenly believe they are slightly different, or worse in direct conflict with one another. This could lead to undesired edits.

I like the idea of having an example or two in the intro, but it’s generally not customary to define something mathematically by giving examples of what it is not, before you adequately defined what it is. If you still want to include an example of what a polynomial is not, then it should be in a separate paragraph, and the definition should stand on its own.

I propose two options.

Option1. Our definition of polynomial in the intro is unique. Most definitions I’ve found, including those in high school text books, say something along the line of:

“A polynomial is an expression involving a sum of one or more variables raised to a power and multiplied by a coefficient. A polynomial in one variable x having coefficients c is given by c_nx^n + c_{n-1}x^{n-1} ... c_2x^2 + c_1x+c_0 \!

I like the idea of using a standard definition that people have likely seen before and are familar with, rather than trying to re-invent the wheel. It’s short and simple and easy for a high school student to understand. When used in the intro, this definition does not need to begin describing what values a power may be, or what kind of numbers the coefficients are, or mention subtraction or division. Those issues can be described in more detail in the Overview. If we try to go much beyond this, trying to cover all the various cases, than we may as well use the definition in the Overview.

Following the definition, add some examples, and then add the last paragraph, “Polynomials are one of the most important concepts ...”

Option2. Have ‘’’one’’’ definition, but create a separate section (called Definition) for it at the top of the page similar to this page: http://en.wikipedia.org/wiki/Orthogonal_polynomials Then take the definition we have in the overview and move it into this new section. —Preceding unsigned comment added by 24.96.130.30 (talk) 07:02, 12 March 2008 (UTC)

Just a few remarks. A mathematical wikipedia article does not need to give a single formal definition; it is not a glossary or a dictionary. And its main vocation is to give infomation about the subject, which is not the limited to descriing how to recognise (in this case) polynomials among other types of mathematical objects. The description in the lead has the difficult task of being understandable to beginners (highschool students) while not offending the specialists by oversimplification. Given the vast amount of applications of polynomials in advanced mathematical subjects, it is natural that the first description should tend more to the simple than to the formal, but it should not oversimplify by for instance mentioning only univariate polynomials. (As an aside, at some point I thought it might be an idea to split off a page "univeriate polynomial" (or instead "multivariate polynomial" is one prefers), so as to separate simpler and more advanced stuff a bit. Just a thought.) This being said, the initial description should be very carefully stated. Personally I think that the fact that some definition is given in a highschool text book does not automatically qualify it for a wikipedia article. Having given a description (not definition) in the intro does not exclude giving a differently worded (and maybe more precise) description later. Of course the relation to the initial description should be pointed out clearly.
Concerning division, I think x2 / 7 is a red herring. It is just an easy way to write down something that involves a fraction, but not a division. For instance, in spite of the fact that 7 is an integer, it is not allowed when considering polynomials with integer coefficients (or as mathemeticians say "over the integers"). On a somewhat similar level one should not be fooled when working out sin(x)2 + x + cos(x)2 to the polynomial x + 1 to think that polynomials may (sometimes) involve sines and cosines. By contrast, subtraction is a completely harmless operation, even though it can always be avoided. So a reader who gets the impression that for polynomials divisions are not allowed while subtractions are, has got an absolutely right impression.
As for you option 1, I doubt that there exists any "standard definition" of polyomials (seriously, I've seen many different approaches, all different and most of them unsuited for this article), but in any case it is an illusion that this would be something most people (particularly high school students) would have seen before. Marc van Leeuwen (talk) 10:17, 12 March 2008 (UTC)

[edit] approximate solutions and exact solutions.

Quote: "As a practical matter, an approximate solution that is accurate to a desired precision may be as useful as an exact solution."

This quote is not really informative. A polynomial defining an exact root is indispensible for analysis, but an approximate value of a root is in many cases much more useful than an exact expression. A root, say 1.61803398875, can be efficiently computed to any desired precision from the polynomial x2−x−1, but it is difficult to reconstruct the polynomial from the approximate value of a root. Bo Jacoby (talk) 13:18, 11 March 2008 (UTC).

[edit] Classification section

I very much disagree with the new "Classification" section. The level of detail there is way out of the proportion to the rest of the article. I would suggest that it be trimmed, if not radically removed.

The way things are now, it takes on my computer five screenfuls to get to such basic stuff like that 1 / (x2 + 1) is not a polynomial, and that the sum of polynomials is a polynomial.

Please keep things in proportion. There are better things to do in the first section than enumerating the polynomials by degree. Oleg Alexandrov (talk) 03:45, 12 March 2008 (UTC)

I don't object to moving the larger table on degrees further down in the article (maybe a separate section?). Doing so may require some minor edits to text were it once was located. I do like what Marc added but some portions of it could be rephrased so they're less verbose. For example, the sentence, "Like for the previous classification, this is about the coefficients one is generally working with; for instance when working with polynomials with complex coefficients one includes polynomials whose coefficients happen to all be real, even though such polynomials can also be considered to be a polynomials with real coefficients." could be shorted and made clearer. —Preceding unsigned comment added by 24.96.130.30 (talk) 06:48, 12 March 2008 (UTC)
I moved the classification section to the bottom, as it stayed in the way of more immediate properties of polynomials. To be honest, I doubt that section has any value, it is just a verbose and uninspired listing of a lot of details, without much significance. Oleg Alexandrov (talk) 05:19, 13 March 2008 (UTC)

[edit] An proposition for a different approach to the lead.

Given the many discussions, old and new, about how to define polynomials. I tend to conclude that there are genuinely different views on this subject. Rather than to propose a definition that suits all tastes, I think it could be better to propose a more pedagogical approach: focus first on how the notion of polynomial arises in mathematics, mention their uses and basic manipulation, and only then give a description of what they are. This has for instance the effect that one can discuss constants and variables, and thier roles, in a natural way first, rather than describing them as part of (or appendix to) the definition of polynomials. One can also prepare for a nuanced view that allows both fairly randomly constructed expressions and neatly arranged sums of terms to cohabit under the designation "polynomial", as in my opinion they should.

Rather than trying some rewrite of the lead to see what comments this gives (I tried this twice this week only to get my edit undone within hours, apparently in neither case because of objections to the edit itself, but somewhat disappinting nonetheless), I'll write my proposition in this talk page, so that people can discuss the pros and cons. Here I go:


In mathematics, polynomials allow computing with one or more values that are not explicitly known, by designating them by letters; they are called variables. Variables can be introduced for various reasons. A variable may represent a fixed but unknown quantity, as it does in the case of polynomial equations; such variables are also called unknowns. This is in fact the oldest use of polynomials. A variable may also designate a quantity that takes many different values at different occasions, as is the case in the definition of a polynomial function; this use is suggested by the name "variable". And a variable may be introduced as an object in itself, distinct from any constant; this is the point of view taken in polynomial rings, and such variables are also called indeterminates. Regardless of the intended application, the rules for manipulating polynomials are the same, and polynomials can be considered without fixing any particular point of view.
A polynomial expression is formed by combining one or more occurences of the variables with each other, and possibly with known values that are called constants, using only addition, subtraction, and multiplication; for instance using the variable x and integer constants, one can form the polynomial expression x(x+3)–8x(x–5)(x+2)–13. The laws of arithmetic, notably the distributive property, allow to work out such expressions to a more regular form, in which multiplication is only applied directly to variables and constants. In our example this form is 8x3 + 25x2 + 83x − 13; here multiplication of several copies of x together is represented as a power of x.
Every polynomial expression can be rewritten as a sum of terms, each of which is the product of at most one constant and zero or more powers of different variables. A polynomial is such a sum of terms; the word polynomial signifies "many terms". In practice the notions of "polynomial expression" and "polynomial" are used interchangeably. It is allowed that none of the terms contains any variable (for instance because all terms containing one cancel out), in which case the polynomial is called a constant polynomial; there might even be no terms at all, which gives the zero polynomial.
Expressions formed using other operations than addition, subtraction, and multiplication, notably those involving division, do not in geneneral lead to a polynomial. For instance the quotient x2 / (x − 3) cannot be rewritten as a sum of terms of the mentioned type, and it does not give a polynomial (it is however an example of the more general notion of a rational function). Similarly, expressions containing exponents other than nonnegative integer numbers, such as x–3, x1/2, or 2x, do not give polynomials, nor do those involving infinite summations.
Polynomials involving only a single variable form the relatively simple but important class of univariate polynomials. They can always be written in the form
c_nx^n+c_{n-1}x^{n-1}+\cdots+c_2x^2+c_1x+c_0,
for some integer n, where cn, …, c1, and c0 denote constants. These constants are called coefficients of the polynomial, and they are allowed to be zero, so that for instance x3–7x can be written as 1x3 + 0x2 − 7x + 0. Except for the case of the zero polynomial, one can however arrange that cn is not zero, by choosing n as small as possible; in this case n is the degree of the polynomial.
Polynomials are one of the most important concepts in algebra and throughout mathematics and science. They can be used to form polynomial equations, which can encode a wide range of problems, from elementary story problems to complicated problems in the sciences; they can be used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics, and are used in calculus polynomials to approximate other functions. Polynomials are also used in their own right to construct polynomial rings, one of the most powerful concepts in algebra and algebraic geometry.


So much for the proposition. I'll give some autocriticism right away. Yes it is (too?) long for a lead. But each paragraph does discuss issues that I consider of prime importance for understanding polynomials (though one may argue that the paragraphs about non-polynomials and about univariate ones could be postponed to after the lead). Some phrases sound a bit clumsy to me (like saying that an expression "gives" a polynomial; I was tempted to say "defines" but that could be misleading to some). I was also tempted to say "nor do in general those involving infinite summations" (because of a long discussion about polynomials and infinite summations), but felt that was probably more confusing than clarifying, and maybe just pedantic. Marc van Leeuwen (talk) 14:43, 12 March 2008 (UTC)

Too long for a lead section. WP:LEAD says a lead section should contain no more than four paragrpahs at most. Two or three paragraphs is best. Gandalf61 (talk) 15:10, 12 March 2008 (UTC)
I agree Gandalf. Too verbose. Oleg Alexandrov (talk) 15:47, 12 March 2008 (UTC)
OK, "too many notes". So which ones to suppress? I'd say how about keeping the first two paragraphs, glue the start of the next paragraph ("Every polynomial expression…") to the previous one (so at least we keep the description of "polynomial"), dropping the part about constant polynomials and the next two paragraphs and keep the final paragraph. That makes three paragraphs in all. If there are still too many words one can start striking out the least useful ones… But I think it kind of useful to keep the example.
Taking a second look I realise that the final paragraph (which was literally taken from the current article) contains quite some redundancy with the initial one, so some reduction can probably be obtained here. Note added by Marc van Leeuwen (talk) 17:44, 12 March 2008 (UTC)
As an aside, I just looked up polynomial in the French Wikipédia, which was quite amusing. It starts (in translation) "In mathematics, a polynomial is a linear combination of products of powers of one or more indeterminates, usually written X, Y, Z, …". At least the know how to get to the point down here (but they didn't get it quite right; I had to insert the "products of" myself). Marc van Leeuwen (talk) 16:24, 12 March 2008 (UTC)

Keep in mind that French schools are better than schools in English speaking countries. Rick Norwood (talk) 12:48, 13 March 2008 (UTC)

Thank you for the compliment (I teach at a French university). What strikes most in comparing French and English language math texts is the sense of formal rigour and abstraction that pervades the French approach. The mentioned French polynomial article says right away that in applied math and analysis polynomials are frequently confused with polynomial functions, but that there is no question of doing that in general (as opposed to elementary) algebra. The related article about construction of a polynomial ring does this over an aribirary, not necessarrily commutative, ring, carefully distinguishing between left and right evaluation of a polynomial at a constant value. You get the point, I think. Fortunately this is not the way we actually teach the subject in France, but the existence of such articles is quite revealing. Marc van Leeuwen (talk) 07:50, 14 March 2008 (UTC)

[edit] A second, shorter, attempt

In mathematics, polynomials allow computing with one or more values that are not explicitly known, by designating them by letters called variables. The oldest use of variables is to represent unknown values, to be deduced from given polynomial equations. Another use, suggested by the name "variable", occurs when defining polynomial functions, where variables designate the varying values at which the function may be evaluated. Finally variables may be introduced formally as indeterminates, new values unrelated to any constants; this leads to the algebraic consstruction of polynomial rings. Regardless of such differences in point of view, the rules for manipulating polynomials are the same.
A polynomial expression is formed by combining one or more occurences of the variables with each other, and possibly with known values that are called constants, using only addition, subtraction, and multiplication; for instance using the variable x and integer constants, one can form the polynomial expression
x(x + 3) − 8x(x − 5)(x + 2) − 13.
The usual laws of arithmetic, notably the distributive property, allow working out such expressions until multiplication is only applied directly to variables and constants; in our example this gives
8x3 + 25x2 + 83x − 13,
with products of several copies of x abbreviated as a powers of x. Thus every polynomial expression can be rewritten as a sum of terms, each of which is the product of a constant and possibly one or more powers of variables. A polynomial is a formal value given by such a sum of terms; the word polynomial signifies "many terms". In practice the notions of "polynomial expression" and "polynomial" are used interchangeably.
Polynomials are one of the most important concepts in mathematics and science. Polynomial equations can encode a wide range of problems, from elementary story problems to complicated problems in the sciences. Polynomial functions appear in settings ranging from basic chemistry and physics to economics, and are form a basic class of functions in calculus, that can be used to approximate more general functions. The construction of polynomial rings is one of the most powerful tools in algebra, notably in algebraic geometry.

Well, the as the sub-heading says, how about this as lead? Marc van Leeuwen (talk) 05:47, 15 March 2008 (UTC)

I find the first paragraph very confusing, unnecessarily detailed, rambling, and not very informative. About the other paragraphs, please try to be short and to the point, you're trying to make fine distinctions which make it hard for the reader to get the general idea. Oleg Alexandrov (talk) 06:32, 15 March 2008 (UTC)
I have now modified the layout of the text so as to clearly display the two (equivalent) examples. I think this corresponds to the "general idea" the reader should get of polynomials, but maybe you meant something else? As for fine distinctions you probably mean the distinction between polynomial expressions and polynomials, which as I stated is usually ignored. But the whole point of my proposition is to mention both where polynomials come from (expressions involving variables formed by addition and multiplications) and how one usually writes them (as a sum of terms), and that this is basically equivalent. I think neutrality requires that both points of view be mentioned; otherwise people will continue fighting and rewriting the definition according to their personal view. But I'll grant that the words "a formal value" can be left out.
As for the "rambling" first paragraph, I merely meant to indicate the occurrences of polynomials to the least initiated (isn't this how articles should start out), sticking to the order in which polynomials enter math instruction (at least mine): first involving unknowns to be solved, then for describing functions, and finally as formal objects by themselves. But to meet your criticism, how about saying "polynomials are expressions that allow" at the beginning (for clarity) and omitting the half-sentence mentioning polynomials rings (to avoid being rambling and too detailed). Marc van Leeuwen (talk) 08:46, 15 March 2008 (UTC)
The first paragraph says more about variables than it does about polynomials. I think you should drop it altogether.
Your example in the second paragraph and your subsequent remarks leave me confused as to how you intend to define a polynomial expression. On the one hand you seem to be saying that a polynomial expression is any expression that can be rewritten as a "a sum of terms, each of which is the product of a constant and possibly one or more powers of variables". On the other hand, you say that a polynomial expression must be formed "using only addition, subtraction, and multiplication". These two definitions are not equivalent. The following expressions
x(x+1) \quad x^2 \left( 1+\frac{1}{x} \right) \quad \frac{x(x^2-1)}{x-1} \quad \sqrt{x^4+2x^3+x^2}
can all be rewritten as x2 + x. Your first definition classifies all of these as polynomial expressions, whereas your second definition only allows the first of these four expressions. Which of these two definitions are you actually proposing ? Gandalf61 (talk) 09:20, 15 March 2008 (UTC)
I think this reply illustrates well why it is good to very carefully formulate the introduction. The text I wrote actually provides answers to most of your questions. Indeed the first paragraph is about variables, not about polynomials. This is intentional and motivated at the very beginning of this talk page section. I think one cannot properly understand polynomials without understanding variables first. My approach was announced as "pedagogical" and I am unashamed to be so; I have noticed that even university students (in France) are quite confused by the distinction between unknowns, variables and indeterminates (not to mention parameters).
As for polyomial expressions, the second paragraph actually defines them, as expressions built up from variables and constants using the mentioned operations. I say that any polynomial expression can be rewritten as a "a sum of terms, each of which is the product of a constant and possibly one or more powers of variables", but if you read carefully I do not say that any expression that can be so rewritten is therefore a polynomial expression; as your examples illustrate, there is no end to the kind of expressions that might reduce to a polynomials (or even to 0). So, except for the first one, your examples are not polynomial expressions, even though they are in some sense equivalent to polynomials (depending on the context, see end of this paragraph). This is one of the reasons why for a precise discussion one should distinguish between the neotion of polynomial expression and polynomial. I did not want to give a definition of polynomial, for which the lead is simply too short, but if you want to know what I really understand by that notion: for me it signifies an element of a polynomial ring, a structure that can be quite precisely defined. (By the way your last three examples do not naturally live in a polynomial ring, although the second and third can be interpreted as rational fractions, in which case they actually turn out to be equal to an element of a subring of polynomials. However, as the body of a function definition they would not be equivalent to a polynomial expression, as they are undefined for some value of x. The last expression by constrast would be equivalent to a polynomial expression mainly if it occurs as definition of a function of a real variable; I cannot think of another context where it would otherwise reduce to x2 + x.) —Preceding unsigned comment added by Marc van Leeuwen (talk • contribs) 12:47, 15 March 2008 (UTC)
I am still somewhat confused about your definition. Are you saying that x(x + 1) is a polynomial because it is a member of a polynomial ring (let us say the ring of polynomials in one variable over the real numbers), whereas \frac{x(x^2-1)}{x-1} is not a member of that ring ? Isn't that like saying that 3x4 is an integer because it is equal to an integer but 24/2 is not an integer because 1/2 is not an integer ? Gandalf61 (talk) 13:33, 15 March 2008 (UTC)
I do not agree that x/x can be rewritten as 1.  --Lambiam 18:24, 15 March 2008 (UTC)
Which shows my point that it depends on the context whether you can rewrite or not. If this is the expression defining a function of a real or complex variable x, then x/x is not equivalent to 1, as it is undefined when x = 0. But working with (formal) polynomials in an indeterminate x, one can observe that in this ring x is exactly divisible by x, and there is a unique quotient 1, so one could say x / x = 1. But the question is whether it makes much sense to write an expression with a division in a setting where exact division is only rarely possible. So I'm not saying that \frac{x(x^2-1)}{x-1} is not a member of the ring \mathbf{R}[x], just that it is not a polynomial expression, and therefore does not automatically designate such a polynomial; rather it is an expression that "accidentally" reduces to a polynomial, just like cos(x)2 + x + sin(x)2 does. The sentence in my proposed first paragraph "Regardless of such differences in point of view, the rules for manipulating polynomials are the same" is there to express that such subtle differences do not arise when one sticks to polynomial expressions: two such expressions describe the same real polynomial function precisely when they describe the same polynomial.
Polynomial rings are constructed by "inventing" new values using variables, just like complex numbers are constructed by "inventing" an imaginary unit i. For such constructions one in general describes two things: what kind of expressions describe arbitrary elements of the structure, and when do two such expressions denote the same value. For rational numbers the expressions are formed as a / b with a,b integer and b nonzero, while a / b = c / d holds if and only if ad = bc. In a similar vein polynomials can be formed by the mentioned "polynomial expressions", and two of them designate the same polynomial if they can be transformed into each other by repeated use of commutatitve, associative and distributive laws. A practical test for that consists of reducing the expressions to sums of products by distributivity, combining similar terms and then comparing coefficients of each monomial. This also shows that we could restrict polynomial expressions to sums of products, without losing any polynomials; yet even then the test for equality is not one of identical expressions. Trying to further restrict the allowed expressions so that polynomials are equal only if they are given by identical expressions is not a very useful exercise (and easy to get wrong). Marc van Leeuwen (talk) 22:49, 15 March 2008 (UTC)
You are again drawing very fine distinctions which confuse more than what they illuminate. I like less and less what you are proposing for this article. Oleg Alexandrov (talk) 05:47, 16 March 2008 (UTC)

The elementary way of explaining polynomials, in my opinion, is the recursive way:

  1. the sum of polynomials is a polynomial
  2. the product of polynomials is a polynomial
  3. any number is a polynomial
  4. the symbol X is a polynomial

If you want to be more exact than this, refer to polynomial ring. Bo Jacoby (talk) 08:47, 16 March 2008 (UTC).

That assumes people have a very good understanding of how recursion generates a set using a starting guess and a set of rules. I doubt that even most college students know this stuff, unless they are computer science majors. So, your idea is elegant, but shifts the burden on understanding a new concept. Oleg Alexandrov (talk) 15:54, 16 March 2008 (UTC)
I think the description by Bo Jacoby is summarized by the phrase "an expression that is constructed from…" in the current lede (this is what I called a polynomial expression), and I think this is not really hard to understand. I agree that this gives an elementary way of explaining polynomials, but I would add two points to the description:
  • A very minor point: one should add that all polynomials are obtained by these rules; as stated they also hold for larger collections like power series
  • More importantly it must be made clear that polynomials are equal if and only if this follows from the commutative, associative and distributive laws, together with arithmetic on constants. As stated without equality criterion, the rules given also apply for instance to complex numbers in place of polynomials, with the complex unit i in place of X and reading "real number" for "number". I will abrreviate the mentioned condition for equality here to "follows from the axioms of commutative rings" (because that is what it is, but I do not think the article should use such terminology).
So for me, the fundamental properties of polynomials that make working with them easy are:
  • Polynomials are precisely the objects that can be designated by polynomial expressions
  • Two polynomials are equal if and only if equality of the expressions giving them follows from the axioms of commutative rings
  • Testing this equality can be done by working out expressions to a sum of products, combining similar terms, and then comparing coefficients
The first two points already describe precisely what is meant by "polynomial", but the third point is what makes life easy, and allows us to imagine general polynomials as sums of terms (thereby also explaining their name). Marc van Leeuwen (talk) 09:39, 17 March 2008 (UTC)
Hmmm ... so your definition of equality of polynomials assumes that polynomials form a commutative ring. Isn't it more usual to define polynomials, equality of polynomials and sums and products of polynomials in a way that is independent of the ring axioms, and then show that they meet these axioms ? This is how it is done in our polynomial ring article.
I think that the "constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication" definition is adequate as a concise and informal definition for the lead, but attempts to expand this into a formal definition seem to run into problems. Gandalf61 (talk) 10:17, 17 March 2008 (UTC)
What I mentioned were properties that to me are fundamental to understanding the "why", "what", and "how" of polynomials. I do not propose them as a formal definition, which could be different and for which polynomial ring is indeed a better place. Marc van Leeuwen (talk) 13:02, 17 March 2008 (UTC)

[edit] Moving easily

Under "Elementary properties" it says after discussing expanded and factored forms of polynomials "In school algebra, students learn to move easily from one form to the other (see: factoring)". This phrase makes me laugh, wondering how well these students will fare for say a random cubic or most any multivariate polynomial. Should we just remove it or try to say something closer to the truth? Marc van Leeuwen (talk) 11:02, 13 March 2008 (UTC)

I would say that the sentence is true in most countries, and true in schools for children of the upper class in England and America. Rick Norwood (talk) 12:50, 13 March 2008 (UTC)
Are you seriously saying there are countries where children (upper class or not) can easily move from the expanded form of a quintic polynomial to its factored form (over the complex numbers)?? I vaguely remember a theorem that claims this should be quite difficult.

How about a modification to the effect that in school mathematics, students learn to move between these two forms in some simple cases? Rick Norwood (talk) 14:58, 14 March 2008 (UTC)

Why mention this at all? Does it have some special significance that makes it notable? For most skills students learn at school, the corresponding articles do not mention this.  --Lambiam 18:21, 15 March 2008 (UTC)

[edit] Hectic polynomials

The adjective "hectic" for a polynomial of degree 100 was recently removed from the article. I don't think the term was just made up - it presumably comes from the same root as the SI prefix hecto-, and there are a couple of independent mentions here and here. Do these qualify as reliable sources, and, if so, should we add "hectic" back into the article (with sources this time) ? Gandalf61 (talk) 09:12, 14 March 2008 (UTC)

I would be interested to know not only whether the derivation of the word "hectic" is correct, but also there is any precedent of anybody calling a polynomial of degree 100 "hectic". And what does one call polynomials of degree 99? Marc van Leeuwen (talk) 13:28, 14 March 2008 (UTC)
I'll argue that having a list of the first ten degree polynomials is enough. I don't think that terminology is used that much anyway. It is much easier to say "polynomial of degree 7" than to look up the corresponding Greek prefix. Oleg Alexandrov (talk) 14:43, 14 March 2008 (UTC)

I first came across the name hectic here [2] , so I included it in the list for completeness. I don't mind removing it, but keep in mind that this is an encyclopedia, and people use it to look up facts on polynomials--even those that may be uncommon. —Preceding unsigned comment added by 24.96.130.30 (talk) 23:45, 16 March 2008 (UTC)

I was glad to learn about "hectic". Rick Norwood (talk) 12:25, 17 March 2008 (UTC)
I am not at all certain that these two references are independent of each other, since they list exactly the same forms, with dual Latin–Greek forms for the 6th and 7th degrees and a Greek-based form for the 100th degree, while the other degrees from 2 upwards are Latin-based. And as one of these (next to Wikipedia) has the only occurrences of "hectic polynomial" on the World Wide Web found by Google search, I am inclined to think this is not only non-notable, but actually a (quite useless) neologism.  --Lambiam 20:48, 17 March 2008 (UTC)

[edit] Rules for multiplying polynomials

In the rule(s) for multiplying polynomials, the text refers to "using the distributive law". If you look at this more formally, there are two kinds of multiplication that play a role here: multiplication of two polynomials, and scalar multiplication of a product of powers of variables by a coefficient, and we need that both kinds distribute over addition – where the latter works as a homomorphism, swapping addition in the ring of polynomials with addiotion in the ring of coefficients, that is, aXk+bXk = (a+b)Xk. Should we just assume that the mathematically less advanced reader does not notice the issues, while the mathematically sophisticate know what is meant anyway?

Alternatively, can't we just say: "using the usual rules of algebra" together with a concrete example, such as

(2x-1)(3x^2+2x+1) = 2x(3x^2+2x+1) - (3x^2+2x+1) = (2x\cdot3x^2+2x\cdot2x+2x\cdot1) - (3x^2+2x+1)
= (6x^3+4x^2+2x)-(3x^2+2x+1) = 6x^3+(4-3)x^2+(2-2)x-1 = 6x^3+x^2-1\,?

I think this has about the same effect (the mathematically less advanced reader does not notice the issues, while the mathematically sophisticate know what is meant anyway), but additionally the likelihood that a reader who does not know this already will actually understand the text is considerably larger.  --Lambiam 20:27, 17 March 2008 (UTC)

I'm currently teaching Abstract Algebra at the graduate level, and the rules for multiplying polynomials are quite complicated, but everybody learns quickly and easily how to do it, because the rule is so similar to the rule for multiplying whole numbers. I think just say "Use the distributive law" and let it go at that. More advanced stuff can come under ring theory. Rick Norwood (talk) 12:53, 18 March 2008 (UTC)

I do not follow. A scalar is also a polynomial, so you need not distinguish between polynomial multiplication and scalar multiplication. The conventional minus notation for negative numbers, however, unnecessarily complicates the rules for polynomial multiplication, and so sign errors are common. The conventional omission of the multiplication sign also makes life difficult for beginners, because you need not distinguish between 2x and 2·x, while you do need to distinguish between 23 and 2·3. Bo Jacoby (talk) 13:07, 18 March 2008 (UTC).
I'd suggest no non-standard dots. Instead of 2x\cdot2x one should write (2x)(2x). That's standard. Oleg Alexandrov (talk) 02:18, 19 March 2008 (UTC)

In giving talks to grade school and high school teachers I have discovered that many (most) do not understand negative numbers, and tell their students that the rules for negative numbers are arbitrary and meaningless. Just memorize them, nobody ever uses them. As we here all know, the rules for negative numbers make sense. When students are taught that negative numbers represent opposties, they stop making so many careless errors. Even when teaching calculus or higer level courses, if a student makes a sign error, I use deposits and withdrawals to correct them. They catch on quickly. But it is difficult to believe how badly negative numbers are taught in American schools.

On the other hand, I like 2x•2x better than (2x)(2x), though either is ok. Note that you can pick up the fat dot at the bottom of this page. Rick Norwood (talk) 12:30, 19 March 2008 (UTC)

I don't quite understand the points raised by Lambiam. The mentioned text occurs in a section "Abstract algebra" and describes a formal construction of polynomials. Looking at things formally, the multiplications and additions that occur inside the expression a_n X^n + a_{n - 1} X^{n - 1} + \cdots + a_1 X + a_0 aren't additions or multiplications at all; they are just part of the notation for a polynomial (in the same sense as the complex number 2 + 3i does not involve an addition; if it did, there would be no way to "perform" that addition). In fact in this approach there isn't really such a thing as X, the best one has is the polynomial 1X + 0. One could say that one accepts aXn as an abbreviation for aX^n+0X^{n-1}+\cdots+0, and that with this convention each polynomial is indeed the sum of the polynomials abbreviated by the terms aiXi occurring in its notation. Only then does it make sense that in multiplying two polynomials the distributive law is applied to reduce to products of the form (aXi)(bXj). I think this is a point where the current text really isn't quite clear; maybe it would be better to denote the objects modeling polynomials less suggestively just as a sequence of coefficients (a_n,a_{n-1},\ldots,a_1,a_0), then no confusion with actual addition and multiplication operators could arise. On the other hand I have the feeling that this would not make for better readability.
One could go even further in this direction and accept Xn as a further abbreviation for 1Xn, and then remark that aXn is indeed equal to (a)(X), that is, to the product of polynomials (aX0)(1Xn). But this is not necessary to interpret the given rule for multiplication (rather, it follows from that rule) so I think it is not worth bothering to introduce an abbreviation just to create ambiguity with the notation used… On the other hand I think article should write the constant term in this context as c0X0 so that the rule for multiplication can apply to it, and it should say that the ring R can then be embedded in R[X] through r\mapsto rX^0. Marc van Leeuwen (talk) 14:29, 19 March 2008 (UTC)

Yes, we need to tell the readers that 0·A=0 and 1·A=A and A+0=A and A0=1 and A1=A and A−B=A+(−1)·B, so that the polynomial 1·x2+(−1)·x1+0·x0 is conventionally written x2−x or x(x−1). Bo Jacoby (talk) 16:06, 19 March 2008 (UTC).

That is in a sense the point. If the scalar a is identified with the polynomial aX0, then of course aX0+bX0 = (a+b)X0 and therefore aXk+bXk = (a+b)Xk. Or is it "of course"? You can't tell whether the operation + in the formula aX0+bX0 is polynomial addition or scalar addition. Identifying the ring in which + operates with a subscript, the idea is apparently that aXk+PbXk = aX0Xk+PbX0Xk = (aX0+PbX0)Xk = (a+Pb)Xk = (a+Sb)X0. But what justifies the step replacing +P by +S? Not only do we need the identification of the scalars with certain polynomials to makes this a valid step, but this also requires the identification to be an embedding, meaning that the algebraic operations in the respective rings commute with the injection, making it a homomorphism. That is not hard to prove after the algebraic operations have been properly defined, but here we are in the process of defining them.  --Lambiam 21:26, 19 March 2008 (UTC) I retract the objection and apologize for not reading the article text more carefully, in which case I should have understood that "Polynomials in X with coefficients in R can be added by simply adding corresponding coefficients" is meant to be a definition of polynomial addition, and not just a factual statement.  --Lambiam 21:35, 19 March 2008 (UTC)

[edit] Who wants to write this?

The Simple English version

Please make it simple.

Lunakeet 12:20, 12 May 2008 (UTC)

I took a shot at it. Needs more work. Later. Rick Norwood (talk) 13:15, 12 May 2008 (UTC)