Talk:Polynomial/Archive 1

message: So far, the polynomial page has special names for degrees up to the 4th. How about past 4th?? Do any of these make sense??
Degree Names from 1 to 12
# Linear
# Quadratic
# Cubic
# Quartic
# Quintic
# Sextic
# Septic
# Octic
# Nonic
# Decic
# Unidecic
# Duodecic
66.32.148.219 00:54, 10 Apr 2004 (UTC)
replies: text: Possibly, but people don't use "special" names for high-degree polynomials. Compare with n-gon. People just say a nth degree polynomial, or the polynomial has degree n. Dysprosia 00:57, 10 Apr 2004 (UTC)

text: The furthest I've heard is "quintic". -- Tarquin 08:08, 10 Apr 2004 (UTC)

text: Sextic, yes. I'd be careful from there on. It may be octavic, for example, for degree 8 sometimes.  Septimic, too. Charles Matthews 08:24, 10 Apr 2004 (UTC)
thread_title:  Degrees past the 4th 

message: This paragraph seems a bit confused - is it talking about computational complexity, or bounding general polynomials in magnitude by their leading term?
Charles Matthews 08:24, 14 Jul 2004 (UTC)
replies: text: I moved the Complexity paragraph to big o notation as an example. MathMartin 16:05, 11 Nov 2004 (UTC)
thread_title: Complexity

message: I think the article is in horrible shape. I have rewritten the definition and restructured the existing material. Some key points of the article should be
#history (finding roots, galois theory)
#numerical analysis
#abstract algebra
MathMartin 21:47, 16 Aug 2004 (UTC)
replies: text: Nice work! :) I agree that overview and history should come first: they are the parts the layperson can understand.-- Tarquin 21:56, 16 Aug 2004 (UTC)
thread_title: Restructuring

message: I think the definition of a (mathematical) term should always be the first subsection of its entry. There must also be an Analysis section under mathematics to which the polynomials entry should be moved. Who's responsible for that?
: It works fine as it is. There may be a way of casually introducing what a polynomial is, before the formal definition sections, however. Dysprosia 01:56, 24 Oct 2004 (UTC)
thread_title:  Definiton and history 

message: Under section Graphs --> Number of x-intercepts,
Earlier erroneous content: For example, a degree 4 polynomial function can have 0, 1, 2, 3 or 4 x-intercepts whereas a degree 5 polynomial function can have 1, 2, 3, 4 or 5 x-intercepts.
Have changed it to: For example, a degree 4 polynomial function can have 0, 2 or 4 x-intercepts whereas a degree 5 polynomial function can have 1, 3 or 5 x-intercepts.
--Wowbagger
:you are correct --MarSch 15:15, 1 November 2005 (UTC)
replies: text: This is not true. Consider the polynomial function y=x^4. This has only 1 x-intercept. Even if you insist it has multiplicity 4, and thus has 4 x intercepts, this is still only true for polynomials with real coefficients. Consider y=x^3(x-sqrt(-1)) This has degree 4 and 1 x-intercept with only multiplicity 3. This should be reverted to say that a polynomial function has x-intercepts (or roots) which number less than or equal to the degree of the polynomial function.Eat2thepieseye 04:31, 11 June 2007 (UTC)
thread_title:  Changed content 

message: I would argue that the introduction of this article was too biased towards numerical analysis, with two paragraphs devoted to that, and there was no mention of abstract polynomials, which are no longer smooth polynomial functions. I think this issue came up earlier, raised by MFH. Anyway, I cut one of the two numerical analysis paragraphs and replaced it with a blurb on abstract algebra. Comments welcome. Oleg Alexandrov (talk) 20:16, 19 December 2005 (UTC)
thread_title:  Reworded intro 

message: "...Laguerre's method which employs complex arithmetic and can locate all complex roots."
This can't be quite right.  Approximate all complex roots?
Also, is there a difference between "total degree" and "degree"?
Answer: The expression xy2 defines several polynomials: x→xy2 of degree 1, y→xy2 of degree 2, and (x,y)→xy2 of degree 3. Bo Jacoby 10:24, 28 February 2006 (UTC)
replies: text: I assume that last is the "total degree".  Thanks. Rick Norwood 14:12, 28 February 2006 (UTC)
thread_title:  questions 

message: If you put your comments at the bottom of the page, they are more likely to be read. That is where people naturally go to look for the newest comments. Rick Norwood 14:11, 6 March 2006 (UTC)
replies: text: Thanks. I'll do that, except for answers, which naturally belong just after the question, I think. Bo Jacoby 15:09, 6 March 2006 (UTC)
thread_title:  Note to Bo Jacoby 

message: The section on roots makes no distinction between exact solutions and approximate solutions.  I'm working on the problem.  In the process, I discovered that algebraic equation also needs a lot of work -- just in case anyone is looking for an important article that cries out for a careful rewrite. Rick Norwood 14:37, 6 March 2006 (UTC)
replies: text: I'm not sure that the distinction needs to be made right here. The Durand-Kerner method seems to provide only approximate solutions, but actually it provides convergent sequences which define exact solutions. So we need not talk about approximations here. Bo Jacoby 15:13, 6 March 2006 (UTC)

text: I disagree with your removing my edit. It was discussed above in the discussion page. Discuss first and edit later please Bo Jacoby 15:19, 6 March 2006 (UTC)
replies: text: The distinction between exact roots and approximate roots is an important one.  Also, in the general article on polynomials, there is no need to repeat technical details found in referenced articles.  The topic of polynomials is a huge one -- it could fill an entire library -- and this article cannot cover everything. Rick Norwood 15:45, 6 March 2006 (UTC)

text: There is but historical reasons to give degrees 2,3 and 4 special treatments. Degree one must have special treatment, division, but all higher degrees are treated by durand-kerner. Solutions in terms of radicals is a dead end. The polynomials define the roots exactly. Computation of radicals is not easier than computation of roots in general. (That is, the equation xn=a is not easier to solve than f(x)=0 where f is a polynomial of degree n). Therefore the distinction between exact roots and approximate roots is not at all important in this context. I agree that the referenced articles should not be repeated, and I did not repeat. You state that solution of equations is difficult. It is no help to the reader to know that it is difficult to you. Don't write articles while you find the subject matter difficult. To me it is not difficult, it is just hard work. I gave examples of how to solve equations, but you deleted it instead of studying it to see that it is not difficult. Your advice to linearize is bad advice in general. You omit the reference to gaussian elimination. The descartes law of signs is of only historical interest. Bo Jacoby 08:11, 7 March 2006 (UTC)
thread_title:  Roots 

message: I've added a small paragraph which gives a graphical explanation to the concept of number of real roots allowed for a polynomial. I want to expand on it a little, but am unsure whether is would serve a useful purpose in the article, or if it would be a waste of time, and a waste of space. Also, I don't think I've explained it clearly... I understand concepts perfectly in my head, but I can't construct them into meaningful descriptions... and i'm not experienced enough with wikipedia to use a diagram to aid the concept (no to mention that it's not really a significant enough concept to warrant a picture in the article. Some feedback would be appreciated. tomohawk 09:51, 18 May 2006 (UTC)
replies: text: I appreciate your effort, but I think this is covered already. Rick Norwood 13:21, 18 May 2006 (UTC)
replies: text: No worries :) tomohawk 13:51, 18 May 2006 (UTC)
thread_title:  Graph of polynomials in cartesian coordinates to understand number of real roots 

message: I'm not sure about the latest change in the lead from
:a polynomial is an expression in which a finite number of constants and variables are combined using only addition, subtraction, multiplication, and non-negative integer exponents (raising to a power).
to
:a polynomial is an expression in which a finite number of constants and variables are combined using only addition and multiplication. Subtraction can be obtained by adding -1 times the subtrahend. Powers with non-negative integral exponents can be obtained by repeated multiplication.
Both are equivalent, but I think the former is simpler for the layman. We could mention the fact later that it can be generated by just addition and multiplication, just not in the lead. --Salix alba (talk) 19:04, 28 September 2006 (UTC)
replies: text: I agree. For now I reverted to the version a month ago which is what you mention first above. Oleg Alexandrov (talk) 02:43, 29 September 2006 (UTC)
replies: text: When I saw the edit by 70.19.63.31, it occurred to me that the definition seems arbitrary. With people throwing in subtraction and some powers, but not other powers. So I thought it would be better to limit the definition to the basics and see the other things as derived properties. After all, if we include subtraction, then why not include division in the definition? JRSpriggs 03:04, 29 September 2006 (UTC)
replies: text: If real numbers can be added, subtracted, and multiplied by, then why can they not be used as exponents? I guess the real issue is what is the motivation for including some functions and not others? JRSpriggs 03:12, 29 September 2006 (UTC)
thread_title:  Only addition and multiplication 

message: The statement: "Analytic solutions of the roots of a polynomial in terms of its coefficients are possible using only the standard arithmetic operations and the extraction of roots only if the degree of the polynomial is four or lower", is not correct. A root of the polynomial x5−a may be expressed using standard arithmetic operations and the extraction of roots, even if the degree is greater than four. (x is a fifth root of a). Please transform the statement into a correct one, or delete it. Bo Jacoby 00:09, 8 December 2006 (UTC)
replies: text: I agree it was not well phrased. I tried to fix that. Oleg Alexandrov (talk) 01:21, 8 December 2006 (UTC)
replies: text: Well done Oleg. Thanks. Bo Jacoby 17:17, 11 December 2006 (UTC)
thread_title:  Roots