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At the high school level quadratic equations are useful in displaying the teacher's facility in proving the quadratic formula, by completing the square. ---- No thanks necessary.----
Questions (I'd rather not make changes to the page since it's out of my area of expertise):
Further examples of polynomials (some are monomials which form a special case with only one term):
If somebody wants to integrate my writeup on E2 to here, feel free. The AC Method may be of particular interest. This is primarily just telling how to factor polynomials so there might be a better place (i.e. factoring) to put it. For simplicity, I'll post a partially wikified version here. If you think it's useful, integrate it. Else, just remove it: http://everything2.org/?node_id=895118 (Note: could contain some errors.)
anxn + an-1xn-1 + an-2xn-2. . . a1x + a0
The degree of a polynomial is the highest total of powers of variables (x, y, etc.) of a single term, so in the polynomial 2xy2 + x2 the degree is three (in the first term, x has a power of one). The standard form of a polynomial is when you write it with the degrees descending (x2 + x + 3, not x + x2 + 3)
To factor a polynomial (If you already know how to then skip down to the AC method. You'll like it. A lot.) you first factor out the common factor, if there is one, using the distributive property:
Ex 1) 2x2 + 4x = 2x(x + 2) Ex 2) 2x2 + 6x + 8 = 2(x2 + 3x + 4)
With a binomial (two terms, as in Ex 1) that's all. If you have a trinomial (three terms, as in Ex 2) you're just getting started.
You usually have to find two binomials (B1 and B2) whose first terms multiply to the first term of your trinomial, last terms multiply to the last term of the trinomial, and B1's first term times B2's last term plus vise versa equals the middle term ( FOIL users: Inside + Outside=Middle)
Ex 3) x2 + 3x + 2 = (x + 1)(x + 2)
If the first term of your trinomial has a coefficient (a) of 1--as shown above--then the first terms of the binomials are x. Otherwise, you have to play around searching for the proper factors to get it right. That's where the following method comes in:
The AC Method
First factor out the common factor. Always, always, always do this.
Ex 4) 6x2 + 2x-4
Now, I know you're thinking, "What if I have a four-term (or more) polynomial?" Easy: Take a few terms, and slap parenthesis around them (Hint, put together terms that have common factors or that look like they'll factor easily.)
Ex 5) 2x3 - 3x2 + 4x - 6
That last example (first and last steps anyway) was taken from College Algebra by Michael Sullivan because I was having a heck of a time making up a good example. (I'm always coming up with prime polynomials in my example and having to modify them so I can factor them. I wish my math teacher had let me do that in my homework.)
Now you need to do some heavy memorising. These are special polynomials and how to factor them. Knowing how to recognise them will help you enormously, both in multiplication and factoring:
Difference of Squares: x2 - a2 = (x - a)(x + a) (Ex 6) x2 - 144 = (x + 12)(x - 12))
Take the coefficients of (x + y)n and look at the nth row of Pascal's Triangle (the "1" at the top is 0th). Cute and useful.
Sorry for the flood. :-)
If this flood might be useful to someone, maybe it belongs on related corrolary pages. stevertigo
Technical point: I've always seen a polynomial defined as an expression, not an equation or a function, ie anx^n + ... + a0. The term "polynomial" is later loosely applied to graphs, functions and equations with a polynomial. -- user:Tarquin
Just wanted to draw everyone's attention to the fact that an anonymous user just changed "In algebra" to "In calculus". with so many mathematicians at Wikipedia, I find it difficult to believe that such an elementary mistake exists in a basic article, and it sounds like something a semi-educated person might think is true. Personally, I haven't the foggiest notion of what calculus is, much less if... calculators? (calculites? calculians?) use polynomials or not. Tokerboy 03:01 Nov 22, 2002 (UTC)
Algebra is a subject. Calculus, on the other hand, is something of a hodge-podge --- a collection of subjects that the curriculum brings together. Algebra goes far beyond those things that most students see, and is a subject to which careers of some researchers are devoted. The topics that go far beyond calculus, on the other hand, are not called "calculus", but go by other names, such as "analysis" and "topology". Therefore, it makes sense to say "in algebra", but not as much to say "in calculus". Polynomials of course appear in calculus, as do many things from algebra. -- Mike Hardy
The reason I separated calculus and algebra is that in algebra, one has to distinguish between polynomials and polynomial functions, while in calculus one doesn't. This point is now lost, in fact the first sentence seems to suggest that the two concepts are the same, which they are only in sloppy calculus usage. AxelBoldt 23:41 Nov 30, 2002 (UTC)
I think this article would be improved if some knowledgeable person would add a few sentences about the Fuchsian Function solution to the paragraph which discusses roots of nth order polynomials. They are hinted at with the existing phrase "degree 5 eluded researchers for a long time", which suggests that a solution was eventually found, but this solution is not mentioned in the article. A new article on Fuchsian Funtions would also be welcome. kielhorn@portland.quik.com Dec 22, 2002
Does anyone know anything about "polynomial arithmetic modulo 2" - you know, the mathematics used for cyclic redundancy checks? Because I don't, and it's not explained in the CRC article, either. -- Tim Starling
Removing this:
Polynomials are surely not the "simplest"; surely f(x) = 0 is "simpler". In addition, something being simple does not imply that it is important. This sentence contributes nothing of value. - Ryguasu 21:11, 13 Sep 2003 (UTC)
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
66.32.148.219 00:54, 10 Apr 2004 (UTC)
Several critics:
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)
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
MathMartin 21:47, 16 Aug 2004 (UTC)
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?
I do not like the latest edits on the definition. Polynomials are a basic topic which should to accessible to a wide range of people. But now the definition will scare off even undergraduate math students. We should have a simple definition which covers the most common cases and terms and then later in the article we can always add the scary stuff for the fearless. MathMartin 16:12, 11 Nov 2004 (UTC)
Yes, the discussion doesn't cut it. Charles Matthews 22:33, 11 Nov 2004 (UTC)
I'll third that. Paul August 22:47, Nov 11, 2004 (UTC)
I reverted the definition to the more simple one I wrote some time ago. Perhaps someone else can integrate the more abstract definition below into the article. MathMartin 13:43, 17 Nov 2004 (UTC)
I admit, the definition below is quite messy (even incomprehensible since I forgot to say what some of the letters used denote) but there should be some kind of general and precise definition of what a polynomial is. Do you want me to have another go in a new paragraph (eg Polynomial, general definition) or try again to integrate it in the existing Defn paragrah? Ncik
Sure, but please don't remove the accessible definition at the top. Perhaps you can integrate your definition into the Abstract algebra paragraph or create a new paragraph Generalization. MathMartin 10:32, 31 Jan 2005 (UTC)
Let us first note that in most cases the term polynomial refers to a term of the following form:
However, this use is, although common, somewhat unprecise since what is actually meant is an univariate polynomial over Q according to the general definition:
Let r, s and t be elements of N, x1,...,xr be variables, F a field and
Furthermore let the finitely many elements of M be denoted by y1,...,yt. Then an r-variate polynomial over F is a term of the form
The ai are called
coefficients. The coefficient of
is called constant coefficient. Polynomials with only one, two or three non-zero coefficients are called monomials, binomials and trinomials, respectively.
The term polynomial can also refer to a function p: M->N, x->p(x), where p(x) is a polynomial as defined above. Such a function may also be called a polynomial function.
The leading coefficient of an univariate polynomial is the coefficient ak which doesn't equal 0 and also has ai=0 for all i > k. We say that a univariate polynomial has degree (or order) k if ak is its leading coefficient, and we say that it is monic or normed if its leading coefficient is 1.
Univariate polynomials of
The question is not whether a definition is "scary" or not, but wether it is a mathematically valid definition or not.
I think we should maintain that "in mathematics" (*sigh*) a polynomial is not a polynomial function.
If we don't at least agree on this, WP becomes useless as a mathematical source of reference.
I say well "agree" and not "write", I mean there can be lots of handwaving and blabla, but (in articles on mathematics) the section entitled "Definition", even if it's at the very end of the article, should really be strictly reserved to a true, mathematical definition on which all textbooks on the subject agree (not only those for primary schools).
I mean, the whole definition, if it's not the true definition, is not worth more than saying "a polynomial is something like x²+5x+3, or 3.9x^7 - 0.01, or any similar expression".
A polynomial is, and will ever remain, a map from N (or some Cartesian power thereof), into a ring (at least), with canonical structure of module and convolution style multiplication. (Or does anybody prefer a terminology involving mysterious abstract "symbols" X which under some obscure conditions can be the same than the symbol 'Y' or even 't', and under some other conditions are different from 'Y' ?)
Once again, I don't mean to explain it like this in the first section, but please, at least allow to mention that there should be a distinction of 'polynomial' from 'polynomial function', even if, by abuse of language, and because on R and C they can be identified, the remainder of the article uses "polynomial" instead of "polynomial function". — MFH: Talk 23:12, 24 May 2005 (UTC)
Perhaps we should change the first sentence to say "In mathematical analysis …" and perhaps refer to the more general notion defined below? Would this help? Paul August ☎ 20:29, May 25, 2005 (UTC)
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
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)
polynomial function redirects to polynomial. But Rational expression redirects to rational function. I don't care which way the redirects go, but it should be consistent. My personal preference is for the object to be the title of the article and the function to be a major subtopic, but I can go either way. I'm placing this comment in the talk pages of both polynomial and rational function in hopes of finding a consensus. Rick Norwood 16:15, 24 December 2005 (UTC)
"...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)
I am not happy with the new section ==Using polynomials for extending the concept of number==. It appears to me like some observation of an amateour mathematician, rather than a serious topic about polynomials belonging in this article. Oleg Alexandrov ( talk) 17:57, 27 February 2006 (UTC)
I could't find the construction of algebraic numbers from natural numbers anywhere else in wikipedia, so I gave it here. The use of polynomials for this purpose is elementary and central, and the idea is standard mathematics and not just 'modestly interesting'. The connection to the theory of ideals needs to be explained further. Tell me the places where you find it badly written so that it can be improved. Spell 'amateur'. :-) Bo Jacoby 09:59, 28 February 2006 (UTC)
Oleg, I took the liberty to change your heading for this section in the discussion page from expressing your emotion and into the subject matter. Bo Jacoby 10:17, 28 February 2006 (UTC)
The construction of the algebraic numbers belongs in algebraic numbers. One "badly written" sentence I noticed was "From this description, addition and multiplication of natural numbers is defined, and the elementary rules of arithmetic are proved." Should be "addition and multiplication...are defined..." There were several others. I don't mean to be too critical -- I did find the section interesting. But it needs to be shorter and more focused, and maybe somewhere else. Number systems might be a good place to start with the Peano axioms, construct negative numbers, pass to the quotient field of the integers, then to polynomials over a field, then to algebraic numbers, and then use Dedikind cuts to get the real numbers and extension fields to get the complex. Then all that is needed here is a reference. Rick Norwood 14:21, 28 February 2006 (UTC)
There is no link to algebraic numbers in the polynomial article. In the sentence "From this description, addition and multiplication of natural numbers is defined", the word "defined" refers to "addition" and "multiplication" (singularis), and not to "natural numbers" (pluralis). I'm glad, Rick, that you found the section interesting. Other readers might find it interesting too. The crux of the peano axioms is that every natural number has a successor. However, I don't approve of Guiseppe Peano's habit of letting the first natural number be 0 rather that 1 (if he was really the one who did it?). Counting to three is saying: "one, two three" and not: "zero, one, two". Thanks, Rick, for the link to number system. The point is that the number extensions to integers, fractions, square roots, and complex, algebraic numbers are defined by polynomial (algebraic) equations. This is a central application of polynomials, which I think belongs in an encyclopedic article on polynomials. I don't mind if the explanations are replaced by references. Please provide references, Oleg, to make your net contribution to this matter become positive rather that negative. Bo Jacoby 08:42, 1 March 2006 (UTC)
Thank you. The idea is the standard one of defining the difference X=a−b by the polynomial equation X+b=a. So manipulation of differences and negative numbers reduce to manipulation with equations of polynomials having natural number coefficients. The same method applies to fractions and to algebraic numbers in general. A reference is number system. I am amazed that you were confused. Please tell me why. Bo Jacoby 14:08, 1 March 2006 (UTC)
To Gandalf: Neither the set of natural numbers, nor the set of polynomials over natural numbers, are rings. The set of integers, and the set of polynomials over integers, are rings. You can define polynomial equations over natural numbers without using (negative) integers, but you cannot define (negative) integers without using polynomial equations over natural numbers, such as X+b=a. So the path of logic has to be: natural numbers → polynomials over natural numbers → integers → polynomials over integers. (That this path is not always followed in number system is perhaps due to history). The discussion on whether zero is considered a natural number is not settled. Most people count "one two three" rather that "zero one two". The additive identity is a formal solution to the polynomial equation X+1=1. I don't have a case to support; I just included this very elementary and very important application of polynomials in the polynomial article. To Oleg: It makes me sad that you are not happy. Wikipedia should be joy and fun. I appreciate that you do a great job for no pay except fun and appreciation, so stop doing it if you don't enjoy it any longer. Our collaboration on complex number and root-finding algorithm resulted in substantial improvements which I found satisfying and I'm sad if you didn't. Thanks for the link to www.linas.org. Bo Jacoby 09:01, 2 March 2006 (UTC)
(1) The polynomials over natural numbers constitute a subset of the ring of polynomials over integers. Only an educated mathematician may get confused. You may also say that 1,2,3,4... are integers, although not every integer occur in the sequence. (2) No, you can not define negative numbers without considering the problem X+b=a. You do not need to hear the word 'polynomial', but the concept is unavoidable. It may be disguised as a riddle like the one quoted in Diophantus. (3) If you change the path, you do break the logic. The later concepts depend on the earlier concepts in the path. (4) That's comforting. However, the habit of quickly removing whatever one dislikes discloses a somewhat tense and impatient emotional state, I think. When Oleg says that he is sorry for being harsh I believe that it is true. Bo Jacoby 15:23, 2 March 2006 (UTC)
The usual construction of integers (reference ?) is that (a,b) is equivalent to (c,d) when the equation X+b=a implies the equation X+d=c, which is iff a+d=b+c (=X+b+d). There is isomorphism between the set of pairs (a,b), and the set of polynomial equations of the form X+b=a. That is not original research but merely explaining the usual construction. The polynomial equation representation motivates the equivalence relation, while the definition a+d=b+c is artificial for the reader. So far the article contains no motivation for the definitions, which is a pity as polynomials are useful for theoretical and practical mathematics. So why does an application of polynomials not belong to this article ? Bo Jacoby 15:00, 6 March 2006 (UTC)
It does not make sense to me to even involve polynomials in the construction of integers. Yes, (a,b) is equivalent to (c,d) when a+d=b+c. There is no need to complicate matters by saying that this is tied to the solution of a polynomial equation. If you do that, you need to construct the polynomials first, which are finite sequences of integers, define their operations, define the concept of root of a polynomial, etc. That would complicate the construction of integers by an order of magnitute. Oleg Alexandrov ( talk) 17:10, 6 March 2006 (UTC)
The definition says: "In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, and multiplication". As constants may be negative, subtraction is actually not needed. Only addition and multiplication is (are?) needed.
The claim: "More powerful methods for solving several polynomial equations in several unknowns are given in linear algebra" does not seem to be true. The article on linear algebra does not contain the promised methods. Do anyone intend to write it? Am I allowed to give it a try ?
The section does not contain examples of polynomials but merely references and should be renamed and moved far away downwards.
The small section "Notes" needs a better name and a better place, perhaps nowhere?
The section "Evaluation of polynomials" fails to define "evaluation". The statement: "Several different algorithms have been developed for this problem. Which algorithm is used for a given polynomial depends on the form of the polynomial and the chosen x" is not helpful and hardly even true. The reader is happier without it.
The sections "Roots" and "Finding roots" should be merged.
Bo Jacoby 12:37, 1 March 2006 (UTC)
Is it "addition and multiplication is defined by induction" or "addition and multiplication are defined by induction" ? When expanding the first version by the distributive law, (a+b)c=ac+bc, you get: "addition is defined by induction and multiplication is defined by induction". So: 'is'. Your example "Monday and Tuesday are days of the week" cannot be expanded into "Monday are days of the week and Tuesday are days of the week" because "days" is (are?) plural. My mothers language was not English, so I am in doubt. Don't hesitate to improve on my English.
Gaussian elimination is limited to the case of linear equations. However the method basicly also applies to the nonlinear case of several polynomial equations with several unknowns. See Buchberger's algorithm and Gröbner basis. Let 0=f(x,y)=g(x,y) be two polynomial equations with two unknowns, x and y. Let A(x,y) be an arbitrary polynomial of x and y. Then the equations 0=f(x,y)+g(x,y)=A(x,y)·f(x,y) follow. So the set (f,g) of polynomials, X(x,y), for which the polynomial equation 0=X(x,y) follow from the original system of equations 0=f(x,y)=g(x,y), is an ideal (ring theory) in the ring Zx][y] of polynomials of y over the ring of polynomials of x. See Greatest_common_divisor#The_gcd_in_commutative_rings. If this ideal, (f,g), contains an element that is of degree zero in y, then y is said to be eliminated and the resulting equation in x alone can be solved numerically by the Durand-Kerner method. The elimination process is: Assume that the degree of f is greater than or equal to the degree of g. (Otherwise switch names). Multiply f and g with monomials so that they have the same leading term, and subtract g from f. Now the degree of f has diminished. Continue until the degree is zero. This algorithm is similar to Euclids algorithm for finding the greatest common divisor between two natural numbers.
Note my edit on number system. Bo Jacoby 10:44, 2 March 2006 (UTC)
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)
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)
Oleg: It does not make sense to me to even involve polynomials in the construction of integers.
Bo: The elementary explanation of subtraction is that a−b solves the polynomial equation X+b=a.
Oleg: (a,b) is equivalent to (c,d) when a+d=b+c. There is no need to complicate matters by saying that this is tied to the solution of a polynomial equation.
Bo: That X+b=a and X+c=d have the same solution, motivates the otherwise ad hoc condition a+d=b+c.
Oleg: If you do that, you need to construct the polynomials first, which are finite sequences of integers,
Bo: Polynomials are defined recursively: "If A,B are polynomials, the so is A+B and AB. 1 is a polynomial, and X is a polynomial".
Oleg: define their operations,
Bo: No new operations, just addition and multiplication and the rules of associativity, commutativity, distributivity and cancellation.
Oleg: define the concept of root of a polynomial,
Bo: only the concept of a solution to an equation.
Oleg: etc.
Bo: no etc, the job is done.
Oleg: That would complicate the construction of integers by an order of magnitude. Oleg Alexandrov (talk) 17:10, 6 March 2006 (UTC)
Bo: Understanding integers is not needed for understanding polynomials. Fewer prerequisites means more readers. Bo Jacoby 09:24, 7 March 2006 (UTC)
Bo Jacoby, OK, so you want to talk serious mathematics. You cannot define polymomials as you say:
This is an intuitive construction, but not a rigurous one. The rigurous construction of polynomials is even more convoluted than the construction of integers from the naturals. That's why it makes no sense to use polynomials to define integers. Oleg Alexandrov ( talk) 17:00, 7 March 2006 (UTC)
Anyone object
Rick, I did not claim that the book on Quantum Mechanics is the place to go for basic mathematical definitions. I merely claim that the reference proves that this is not original research on my part. It is true that N[X] is not a ring, but it is not to the point, for we don't need a ring. The extensions of numbers N→Z→Q match the extensions of polynomials N[X]→Z[X]→Q[X].
MFH, The first example given on Polynomial is in Z[X] but not in N[X]. The elements of N[X] are polynomials just as the elements of N are numbers.
Oleg, 'my' definition compares favorably to all the explanations in the article. If you or MFH know a rigorous (not rigurous) definition which is also comprehensable to the readers, how come it does not appear in the article ? Bo Jacoby 12:00, 8 March 2006 (UTC)
You are right and I am embarassed. Bo Jacoby 12:47, 8 March 2006 (UTC)
I wrote what I mean and I mean what I wrote: "the same solution". Equations having no solution do not have the same solution, even if they do have the same (empty) set of solutions. However, the relation as defined is symmetric and transitive, but not reflexive, so I must specify that R1 is the reflexive closure. Thank you. Bo Jacoby 14:45, 8 March 2006 (UTC)
Bo, all this argument is because we did not go to the bottom of things. Let me ask you:
Also, what is an indeterminate, X? You are avoiding this question, and you are trying to construct the integers using polynomials, which were not defined. You must construct polynomials first, please understand that. You can't just say let X be an indeterminate. Oleg Alexandrov ( talk) 15:59, 8 March 2006 (UTC)
Oleg - The set, N[X], of polynomials is defined recursively, just as the set, N, of natural numbers is defined recursively by Peanos axioms. You are not told what a natural number is, only that the axioms apply. Basicly the axioms says that if a is in N then so is a+1, and that 1 is in N. You are not told what 1 is, and you are not told what +1 means. From the axioms, addition and multiplication are defined, and the arithmetic rules, a+b=b+a. (a+b)+c=a+(b+c), a(b+c)=ab+ac, ab=ba, (ab)c=a(bc), a+b=a+c => b=c, and ab=ac => b=c are proved. N is closed under addition and multiplication: If a,b are in N then so is a+b and ab. N is characterized by these two conditions. (1): N is closed under addition and multiplication, and (2): 1 is in N. Now define N[X] by similar conditions: (1): N[X] is closed under addition and multiplication, (2): 1 is in N[X], and (3): X is in N[X]. You are not told what X is, only that the rules apply. That is a perfectly sound definition. A solution to an equation, f(X)=g(X), is a number, a, which substituted for X in the equation produces a true statement: f(a)=g(a). A solution is not a set of numbers, but a number. The equation X+2=0 has no solution in N. Nor has the equations X+3=0 a solution in N. So they do not have the same solution. The two equations do not have different solutions. They have no solutions.
Gandalf - No, I do not want X+2=1 (A) and X+3=2 (C) to be in the same equivalence class under the relation R1 , as the two equations do not have the same solution in N. But they are equivalent under the extended relation R2 because 2+2=1+3. So the two equations define the same integer, which is the solution to either in Z.
Rick - It is not nonsens what I wrote about long division. I explained the difference between exact and approximate solutions to polynomial equations, the point being that the root of a first degree polynomial like 3x-1 cannot be expressed exactly any more or any less than the roots of a fifth degree polynomial like x5-x-1. The expression x=1/3 says nothing more than 3x-1=0. I you want to know the root you must approximate: x=0.3333. This is done by long division. Your claim: "When the polynomial is of degree greater than four, it is not always possible to find exact expressions for the zeroes" is not true, because the polynomial itself is an exact expression for the roots. The descartes method is basicly useless and has but historical interest and does not belong in an introduction. Please hesitate reverting other peoples edit. Be open minded and don't reject new stuff until you understand it. The article is very bad by now. There is no proper definition, the history section is very narrow, there are no application, and a lot of the text is untrue. It need to be improved, but it cannot be improved when you just revert any change. I don't expect you to agree, but you should ask questions first and shoot afterwards. Bo Jacoby 01:54, 9 March 2006 (UTC)
You wrote:
Look, you are saying it is false that the equations have the same solution, and it is also false that they don't have the same solution. That can't be, logically. Oleg Alexandrov ( talk) 03:03, 9 March 2006 (UTC)
They have the same solution set (the empty set), which I think is really what matters here. Dysprosia 04:07, 9 March 2006 (UTC)
It is logical to say that "you and I are neither dating the same girl nor different girls" if at least one of us is not dating any girl. Now substitute "dating a girl" with "having a solution". A solution is not the same thing as the set of solutions - if the set of solutions is empty, then there is no solution. The relationship (X+a=b)R1(X+c=d) does not mean that the two equations have the same set of solutions, but the relationship is true EITHER if each of the two equations has a solution and these solutions are the same, OR if a=c and b=d. (Any equation must be equivalent to itself). This relationship implies that a+d=b+c, that is: (X+a=b)R2(X+c=d). This implication does not go backwards: The equivalence relation R2 is a proper extension to R1. It contains more relationships. Let Z=M/R2 be the new set of equivalence classes of equations. There is a one-to-one correspondance between N and the subset of Z of equivalence classes of equations having the same solutions in N . A number is identified with the equivalence class of equation having that number as a solution. This identification extends the concept of number from N to Z. Bo Jacoby 10:01, 9 March 2006 (UTC)
Thanks for your contributions so far. How do you explain b−a with a,b in N ? To me it is a solution to the equation X+a=b. (Postulating R2 without R1 would omit this explanation). The same approach explains fractions and radicals and algebraic numbers. What explanation is natural to you ? It is your turn to be constructive. I'll then resist the temptation of being too critical. :) Bo Jacoby 11:19, 9 March 2006 (UTC)
You are definining a set M={X+b=a | a,b in N}. You can't do that. You cannot define a set of equalities, an equality is not a set element. You should say the set of all pairs (X+b, a), now that's correct. Oleg Alexandrov ( talk) 16:02, 9 March 2006 (UTC)
Rick - Every child in the whole world has to learn what an integer is, so it must be well explained in wikipedia. By now it is not well explained. My approach obviously provokes not only a lot of constructiv mathematical discussion but also some impatient dogmatic opposition. So I challenge my honored opponents to write an explanation which is understandable to the child and acceptable to Oleg and even to myself. Do you think that is easy ?
Oleg - You agree to consider sets of numbers, sets of pairs of numbers, and even sets of pairs of polynomials, but you disagree to consider sets of equations. Why is that ? Isn't an equation a mathematical object too ? The math teacher writes on the blackboard: , points at it, and says: "This is an equation". An equation, like a polynomial, has several interpretations: unevaluated it is a formal mathematical object - fully evaluated it is a truth value. The equation X+a=b is unevaluated - it is not a claim that the two sides are equal polynomials. The polynomial "X+a" is also unevaluated - It is not a number, but either a function, x→(x+a), or an abstract mathematical object in its own right. The equation X+a=b may be considered a boolean function of an integer: x→(x+a=b). Here it even means (a,b)→(x→(x+a=b)) : to every pair, (a,b), there correspond a function that to a number x, tests whether x+a=b is true or not. If we stop talking about these equations and just talk about pairs then nobody will understand the meaning of the equivalence a+d=b+c. Your objection indicates that we should write fully: M = { (a,b)→(x→(x+a=b)) | a,b,x in N } rather than shorthand: M = { X+a=b | a,b in N } . Personally I prefer using leftarrow rather that rightarrow for specifying functions, f=(y←x) meaning y=f(x), because y is more important that x, but that is nonstandard. In that notation the definition would be like this: M = { ((x+a=b)←x)←(a,b) | a,b,x in N } Bo Jacoby 10:22, 10 March 2006 (UTC)
If you are trying to explain something to others, like above, you should use common notation, not your own notation. :) But I think I got it. Thanks.
Look, you are saying that the equations X+3=1 and X+4=2 define the same solution in N. That is wrong. Assume that the solution is the same, call it n≥0. By plugging it in the first equation you see that it does not satisfy the equation, contradiction. Oleg Alexandrov ( talk) 16:24, 10 March 2006 (UTC)
Oleg, I agree that the equations X+3=1 and X+4=2 do not have the same solution in N. Neither of the two equations have a solution in N. So the two equations are not equivalent under the relation R1 (which demands that two equations have the same solution in N). But they are equivalent under the extended equivalence relation R2, because 3+2=1+4. So they define the same element in Z. Remember that the Z-element b−a is the set of equations of the form X+c=d where a+d=b+c. Dysprosia's last remark is perfectly correct. Bo Jacoby 23:15, 13 March 2006 (UTC)
Oleg, going directly to R2, the approach cannot be reused for constructing fractions and algebraic numbers. I was not trying to get to both the intuitive motivation and the rigorous construction in one shot. I wrote the intuitive non-rigorous section
You then requested a rigorous presentation, claiming that it couldn't be made. I showed that it can be made, albeit confusing, as rigor often is.
Dysprosia, If you have a construction that is really standard, lucid, verifiable, and source-able, then write it down and let us use it. Let us not solve a solved problem. Bo Jacoby 13:42, 14 March 2006 (UTC)
I enjoyed the discussion. When your arguments were refuted, you did not change your mind regarding the conclusion, but you just invented new objections. So your argumentation is reverse, constructing arguments for your conclusion rather than deriving a conclusion from the arguments. Surely, that kind of argumentation can go on till we drop. My purpose is not to make you change your mind, because that probably can not be done anyway, but to improve the formulation. You did pinpoint weak spots in the text. By now the polynomial article is without motivation and application. That is bad. Also there is nowhere an explanation for the construction of integers, fractions and algebraic numbers in general. That is bad too. I trust that you guys will mend these defects, as you do not allow me to do it. Bo Jacoby 09:02, 15 March 2006 (UTC)
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)
Polynomials are important for a number of reasons: 1. A sum of polynomials is a polynomial 2. A product of polynomials is a polynomial 3. The derivative of a polynomial is a polynomial 4. The integral of a polynomial is a polynomial 5. Polynomials serve to approximate many functions, such as sine, cosine, and exponential.
This text from www.jsoftware.com contains motivation in a nutshell. There is presently no section on motivation in the article. Bo Jacoby 11:04, 22 June 2006 (UTC)
I'm not sure about the latest change in the lead from
to
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)
We don't get to decide what we would like the word "polynomial" to mean. The word has a well established meaning, and that meaning does not allow non-natural real number powers or division by a variable. Rick Norwood 13:24, 9 December 2006 (UTC)
The anons recent edit to the lead raises an interesting distinction between a purely formal definition of a polynomial as the sum of monomials, and equations which can be reduced to that form. For example Stewart Galois Theory defines polynomials as a formal sum. Hence the formal expression is not a polynomial but its expanded form is. Should we make this clear in the lead? -- Salix alba ( talk) 01:03, 5 November 2006 (UTC)
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)
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 | Archive 3 | Archive 4 |
At the high school level quadratic equations are useful in displaying the teacher's facility in proving the quadratic formula, by completing the square. ---- No thanks necessary.----
Questions (I'd rather not make changes to the page since it's out of my area of expertise):
Further examples of polynomials (some are monomials which form a special case with only one term):
If somebody wants to integrate my writeup on E2 to here, feel free. The AC Method may be of particular interest. This is primarily just telling how to factor polynomials so there might be a better place (i.e. factoring) to put it. For simplicity, I'll post a partially wikified version here. If you think it's useful, integrate it. Else, just remove it: http://everything2.org/?node_id=895118 (Note: could contain some errors.)
anxn + an-1xn-1 + an-2xn-2. . . a1x + a0
The degree of a polynomial is the highest total of powers of variables (x, y, etc.) of a single term, so in the polynomial 2xy2 + x2 the degree is three (in the first term, x has a power of one). The standard form of a polynomial is when you write it with the degrees descending (x2 + x + 3, not x + x2 + 3)
To factor a polynomial (If you already know how to then skip down to the AC method. You'll like it. A lot.) you first factor out the common factor, if there is one, using the distributive property:
Ex 1) 2x2 + 4x = 2x(x + 2) Ex 2) 2x2 + 6x + 8 = 2(x2 + 3x + 4)
With a binomial (two terms, as in Ex 1) that's all. If you have a trinomial (three terms, as in Ex 2) you're just getting started.
You usually have to find two binomials (B1 and B2) whose first terms multiply to the first term of your trinomial, last terms multiply to the last term of the trinomial, and B1's first term times B2's last term plus vise versa equals the middle term ( FOIL users: Inside + Outside=Middle)
Ex 3) x2 + 3x + 2 = (x + 1)(x + 2)
If the first term of your trinomial has a coefficient (a) of 1--as shown above--then the first terms of the binomials are x. Otherwise, you have to play around searching for the proper factors to get it right. That's where the following method comes in:
The AC Method
First factor out the common factor. Always, always, always do this.
Ex 4) 6x2 + 2x-4
Now, I know you're thinking, "What if I have a four-term (or more) polynomial?" Easy: Take a few terms, and slap parenthesis around them (Hint, put together terms that have common factors or that look like they'll factor easily.)
Ex 5) 2x3 - 3x2 + 4x - 6
That last example (first and last steps anyway) was taken from College Algebra by Michael Sullivan because I was having a heck of a time making up a good example. (I'm always coming up with prime polynomials in my example and having to modify them so I can factor them. I wish my math teacher had let me do that in my homework.)
Now you need to do some heavy memorising. These are special polynomials and how to factor them. Knowing how to recognise them will help you enormously, both in multiplication and factoring:
Difference of Squares: x2 - a2 = (x - a)(x + a) (Ex 6) x2 - 144 = (x + 12)(x - 12))
Take the coefficients of (x + y)n and look at the nth row of Pascal's Triangle (the "1" at the top is 0th). Cute and useful.
Sorry for the flood. :-)
If this flood might be useful to someone, maybe it belongs on related corrolary pages. stevertigo
Technical point: I've always seen a polynomial defined as an expression, not an equation or a function, ie anx^n + ... + a0. The term "polynomial" is later loosely applied to graphs, functions and equations with a polynomial. -- user:Tarquin
Just wanted to draw everyone's attention to the fact that an anonymous user just changed "In algebra" to "In calculus". with so many mathematicians at Wikipedia, I find it difficult to believe that such an elementary mistake exists in a basic article, and it sounds like something a semi-educated person might think is true. Personally, I haven't the foggiest notion of what calculus is, much less if... calculators? (calculites? calculians?) use polynomials or not. Tokerboy 03:01 Nov 22, 2002 (UTC)
Algebra is a subject. Calculus, on the other hand, is something of a hodge-podge --- a collection of subjects that the curriculum brings together. Algebra goes far beyond those things that most students see, and is a subject to which careers of some researchers are devoted. The topics that go far beyond calculus, on the other hand, are not called "calculus", but go by other names, such as "analysis" and "topology". Therefore, it makes sense to say "in algebra", but not as much to say "in calculus". Polynomials of course appear in calculus, as do many things from algebra. -- Mike Hardy
The reason I separated calculus and algebra is that in algebra, one has to distinguish between polynomials and polynomial functions, while in calculus one doesn't. This point is now lost, in fact the first sentence seems to suggest that the two concepts are the same, which they are only in sloppy calculus usage. AxelBoldt 23:41 Nov 30, 2002 (UTC)
I think this article would be improved if some knowledgeable person would add a few sentences about the Fuchsian Function solution to the paragraph which discusses roots of nth order polynomials. They are hinted at with the existing phrase "degree 5 eluded researchers for a long time", which suggests that a solution was eventually found, but this solution is not mentioned in the article. A new article on Fuchsian Funtions would also be welcome. kielhorn@portland.quik.com Dec 22, 2002
Does anyone know anything about "polynomial arithmetic modulo 2" - you know, the mathematics used for cyclic redundancy checks? Because I don't, and it's not explained in the CRC article, either. -- Tim Starling
Removing this:
Polynomials are surely not the "simplest"; surely f(x) = 0 is "simpler". In addition, something being simple does not imply that it is important. This sentence contributes nothing of value. - Ryguasu 21:11, 13 Sep 2003 (UTC)
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
66.32.148.219 00:54, 10 Apr 2004 (UTC)
Several critics:
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)
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
MathMartin 21:47, 16 Aug 2004 (UTC)
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?
I do not like the latest edits on the definition. Polynomials are a basic topic which should to accessible to a wide range of people. But now the definition will scare off even undergraduate math students. We should have a simple definition which covers the most common cases and terms and then later in the article we can always add the scary stuff for the fearless. MathMartin 16:12, 11 Nov 2004 (UTC)
Yes, the discussion doesn't cut it. Charles Matthews 22:33, 11 Nov 2004 (UTC)
I'll third that. Paul August 22:47, Nov 11, 2004 (UTC)
I reverted the definition to the more simple one I wrote some time ago. Perhaps someone else can integrate the more abstract definition below into the article. MathMartin 13:43, 17 Nov 2004 (UTC)
I admit, the definition below is quite messy (even incomprehensible since I forgot to say what some of the letters used denote) but there should be some kind of general and precise definition of what a polynomial is. Do you want me to have another go in a new paragraph (eg Polynomial, general definition) or try again to integrate it in the existing Defn paragrah? Ncik
Sure, but please don't remove the accessible definition at the top. Perhaps you can integrate your definition into the Abstract algebra paragraph or create a new paragraph Generalization. MathMartin 10:32, 31 Jan 2005 (UTC)
Let us first note that in most cases the term polynomial refers to a term of the following form:
However, this use is, although common, somewhat unprecise since what is actually meant is an univariate polynomial over Q according to the general definition:
Let r, s and t be elements of N, x1,...,xr be variables, F a field and
Furthermore let the finitely many elements of M be denoted by y1,...,yt. Then an r-variate polynomial over F is a term of the form
The ai are called
coefficients. The coefficient of
is called constant coefficient. Polynomials with only one, two or three non-zero coefficients are called monomials, binomials and trinomials, respectively.
The term polynomial can also refer to a function p: M->N, x->p(x), where p(x) is a polynomial as defined above. Such a function may also be called a polynomial function.
The leading coefficient of an univariate polynomial is the coefficient ak which doesn't equal 0 and also has ai=0 for all i > k. We say that a univariate polynomial has degree (or order) k if ak is its leading coefficient, and we say that it is monic or normed if its leading coefficient is 1.
Univariate polynomials of
The question is not whether a definition is "scary" or not, but wether it is a mathematically valid definition or not.
I think we should maintain that "in mathematics" (*sigh*) a polynomial is not a polynomial function.
If we don't at least agree on this, WP becomes useless as a mathematical source of reference.
I say well "agree" and not "write", I mean there can be lots of handwaving and blabla, but (in articles on mathematics) the section entitled "Definition", even if it's at the very end of the article, should really be strictly reserved to a true, mathematical definition on which all textbooks on the subject agree (not only those for primary schools).
I mean, the whole definition, if it's not the true definition, is not worth more than saying "a polynomial is something like x²+5x+3, or 3.9x^7 - 0.01, or any similar expression".
A polynomial is, and will ever remain, a map from N (or some Cartesian power thereof), into a ring (at least), with canonical structure of module and convolution style multiplication. (Or does anybody prefer a terminology involving mysterious abstract "symbols" X which under some obscure conditions can be the same than the symbol 'Y' or even 't', and under some other conditions are different from 'Y' ?)
Once again, I don't mean to explain it like this in the first section, but please, at least allow to mention that there should be a distinction of 'polynomial' from 'polynomial function', even if, by abuse of language, and because on R and C they can be identified, the remainder of the article uses "polynomial" instead of "polynomial function". — MFH: Talk 23:12, 24 May 2005 (UTC)
Perhaps we should change the first sentence to say "In mathematical analysis …" and perhaps refer to the more general notion defined below? Would this help? Paul August ☎ 20:29, May 25, 2005 (UTC)
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
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)
polynomial function redirects to polynomial. But Rational expression redirects to rational function. I don't care which way the redirects go, but it should be consistent. My personal preference is for the object to be the title of the article and the function to be a major subtopic, but I can go either way. I'm placing this comment in the talk pages of both polynomial and rational function in hopes of finding a consensus. Rick Norwood 16:15, 24 December 2005 (UTC)
"...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)
I am not happy with the new section ==Using polynomials for extending the concept of number==. It appears to me like some observation of an amateour mathematician, rather than a serious topic about polynomials belonging in this article. Oleg Alexandrov ( talk) 17:57, 27 February 2006 (UTC)
I could't find the construction of algebraic numbers from natural numbers anywhere else in wikipedia, so I gave it here. The use of polynomials for this purpose is elementary and central, and the idea is standard mathematics and not just 'modestly interesting'. The connection to the theory of ideals needs to be explained further. Tell me the places where you find it badly written so that it can be improved. Spell 'amateur'. :-) Bo Jacoby 09:59, 28 February 2006 (UTC)
Oleg, I took the liberty to change your heading for this section in the discussion page from expressing your emotion and into the subject matter. Bo Jacoby 10:17, 28 February 2006 (UTC)
The construction of the algebraic numbers belongs in algebraic numbers. One "badly written" sentence I noticed was "From this description, addition and multiplication of natural numbers is defined, and the elementary rules of arithmetic are proved." Should be "addition and multiplication...are defined..." There were several others. I don't mean to be too critical -- I did find the section interesting. But it needs to be shorter and more focused, and maybe somewhere else. Number systems might be a good place to start with the Peano axioms, construct negative numbers, pass to the quotient field of the integers, then to polynomials over a field, then to algebraic numbers, and then use Dedikind cuts to get the real numbers and extension fields to get the complex. Then all that is needed here is a reference. Rick Norwood 14:21, 28 February 2006 (UTC)
There is no link to algebraic numbers in the polynomial article. In the sentence "From this description, addition and multiplication of natural numbers is defined", the word "defined" refers to "addition" and "multiplication" (singularis), and not to "natural numbers" (pluralis). I'm glad, Rick, that you found the section interesting. Other readers might find it interesting too. The crux of the peano axioms is that every natural number has a successor. However, I don't approve of Guiseppe Peano's habit of letting the first natural number be 0 rather that 1 (if he was really the one who did it?). Counting to three is saying: "one, two three" and not: "zero, one, two". Thanks, Rick, for the link to number system. The point is that the number extensions to integers, fractions, square roots, and complex, algebraic numbers are defined by polynomial (algebraic) equations. This is a central application of polynomials, which I think belongs in an encyclopedic article on polynomials. I don't mind if the explanations are replaced by references. Please provide references, Oleg, to make your net contribution to this matter become positive rather that negative. Bo Jacoby 08:42, 1 March 2006 (UTC)
Thank you. The idea is the standard one of defining the difference X=a−b by the polynomial equation X+b=a. So manipulation of differences and negative numbers reduce to manipulation with equations of polynomials having natural number coefficients. The same method applies to fractions and to algebraic numbers in general. A reference is number system. I am amazed that you were confused. Please tell me why. Bo Jacoby 14:08, 1 March 2006 (UTC)
To Gandalf: Neither the set of natural numbers, nor the set of polynomials over natural numbers, are rings. The set of integers, and the set of polynomials over integers, are rings. You can define polynomial equations over natural numbers without using (negative) integers, but you cannot define (negative) integers without using polynomial equations over natural numbers, such as X+b=a. So the path of logic has to be: natural numbers → polynomials over natural numbers → integers → polynomials over integers. (That this path is not always followed in number system is perhaps due to history). The discussion on whether zero is considered a natural number is not settled. Most people count "one two three" rather that "zero one two". The additive identity is a formal solution to the polynomial equation X+1=1. I don't have a case to support; I just included this very elementary and very important application of polynomials in the polynomial article. To Oleg: It makes me sad that you are not happy. Wikipedia should be joy and fun. I appreciate that you do a great job for no pay except fun and appreciation, so stop doing it if you don't enjoy it any longer. Our collaboration on complex number and root-finding algorithm resulted in substantial improvements which I found satisfying and I'm sad if you didn't. Thanks for the link to www.linas.org. Bo Jacoby 09:01, 2 March 2006 (UTC)
(1) The polynomials over natural numbers constitute a subset of the ring of polynomials over integers. Only an educated mathematician may get confused. You may also say that 1,2,3,4... are integers, although not every integer occur in the sequence. (2) No, you can not define negative numbers without considering the problem X+b=a. You do not need to hear the word 'polynomial', but the concept is unavoidable. It may be disguised as a riddle like the one quoted in Diophantus. (3) If you change the path, you do break the logic. The later concepts depend on the earlier concepts in the path. (4) That's comforting. However, the habit of quickly removing whatever one dislikes discloses a somewhat tense and impatient emotional state, I think. When Oleg says that he is sorry for being harsh I believe that it is true. Bo Jacoby 15:23, 2 March 2006 (UTC)
The usual construction of integers (reference ?) is that (a,b) is equivalent to (c,d) when the equation X+b=a implies the equation X+d=c, which is iff a+d=b+c (=X+b+d). There is isomorphism between the set of pairs (a,b), and the set of polynomial equations of the form X+b=a. That is not original research but merely explaining the usual construction. The polynomial equation representation motivates the equivalence relation, while the definition a+d=b+c is artificial for the reader. So far the article contains no motivation for the definitions, which is a pity as polynomials are useful for theoretical and practical mathematics. So why does an application of polynomials not belong to this article ? Bo Jacoby 15:00, 6 March 2006 (UTC)
It does not make sense to me to even involve polynomials in the construction of integers. Yes, (a,b) is equivalent to (c,d) when a+d=b+c. There is no need to complicate matters by saying that this is tied to the solution of a polynomial equation. If you do that, you need to construct the polynomials first, which are finite sequences of integers, define their operations, define the concept of root of a polynomial, etc. That would complicate the construction of integers by an order of magnitute. Oleg Alexandrov ( talk) 17:10, 6 March 2006 (UTC)
The definition says: "In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, and multiplication". As constants may be negative, subtraction is actually not needed. Only addition and multiplication is (are?) needed.
The claim: "More powerful methods for solving several polynomial equations in several unknowns are given in linear algebra" does not seem to be true. The article on linear algebra does not contain the promised methods. Do anyone intend to write it? Am I allowed to give it a try ?
The section does not contain examples of polynomials but merely references and should be renamed and moved far away downwards.
The small section "Notes" needs a better name and a better place, perhaps nowhere?
The section "Evaluation of polynomials" fails to define "evaluation". The statement: "Several different algorithms have been developed for this problem. Which algorithm is used for a given polynomial depends on the form of the polynomial and the chosen x" is not helpful and hardly even true. The reader is happier without it.
The sections "Roots" and "Finding roots" should be merged.
Bo Jacoby 12:37, 1 March 2006 (UTC)
Is it "addition and multiplication is defined by induction" or "addition and multiplication are defined by induction" ? When expanding the first version by the distributive law, (a+b)c=ac+bc, you get: "addition is defined by induction and multiplication is defined by induction". So: 'is'. Your example "Monday and Tuesday are days of the week" cannot be expanded into "Monday are days of the week and Tuesday are days of the week" because "days" is (are?) plural. My mothers language was not English, so I am in doubt. Don't hesitate to improve on my English.
Gaussian elimination is limited to the case of linear equations. However the method basicly also applies to the nonlinear case of several polynomial equations with several unknowns. See Buchberger's algorithm and Gröbner basis. Let 0=f(x,y)=g(x,y) be two polynomial equations with two unknowns, x and y. Let A(x,y) be an arbitrary polynomial of x and y. Then the equations 0=f(x,y)+g(x,y)=A(x,y)·f(x,y) follow. So the set (f,g) of polynomials, X(x,y), for which the polynomial equation 0=X(x,y) follow from the original system of equations 0=f(x,y)=g(x,y), is an ideal (ring theory) in the ring Zx][y] of polynomials of y over the ring of polynomials of x. See Greatest_common_divisor#The_gcd_in_commutative_rings. If this ideal, (f,g), contains an element that is of degree zero in y, then y is said to be eliminated and the resulting equation in x alone can be solved numerically by the Durand-Kerner method. The elimination process is: Assume that the degree of f is greater than or equal to the degree of g. (Otherwise switch names). Multiply f and g with monomials so that they have the same leading term, and subtract g from f. Now the degree of f has diminished. Continue until the degree is zero. This algorithm is similar to Euclids algorithm for finding the greatest common divisor between two natural numbers.
Note my edit on number system. Bo Jacoby 10:44, 2 March 2006 (UTC)
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)
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)
Oleg: It does not make sense to me to even involve polynomials in the construction of integers.
Bo: The elementary explanation of subtraction is that a−b solves the polynomial equation X+b=a.
Oleg: (a,b) is equivalent to (c,d) when a+d=b+c. There is no need to complicate matters by saying that this is tied to the solution of a polynomial equation.
Bo: That X+b=a and X+c=d have the same solution, motivates the otherwise ad hoc condition a+d=b+c.
Oleg: If you do that, you need to construct the polynomials first, which are finite sequences of integers,
Bo: Polynomials are defined recursively: "If A,B are polynomials, the so is A+B and AB. 1 is a polynomial, and X is a polynomial".
Oleg: define their operations,
Bo: No new operations, just addition and multiplication and the rules of associativity, commutativity, distributivity and cancellation.
Oleg: define the concept of root of a polynomial,
Bo: only the concept of a solution to an equation.
Oleg: etc.
Bo: no etc, the job is done.
Oleg: That would complicate the construction of integers by an order of magnitude. Oleg Alexandrov (talk) 17:10, 6 March 2006 (UTC)
Bo: Understanding integers is not needed for understanding polynomials. Fewer prerequisites means more readers. Bo Jacoby 09:24, 7 March 2006 (UTC)
Bo Jacoby, OK, so you want to talk serious mathematics. You cannot define polymomials as you say:
This is an intuitive construction, but not a rigurous one. The rigurous construction of polynomials is even more convoluted than the construction of integers from the naturals. That's why it makes no sense to use polynomials to define integers. Oleg Alexandrov ( talk) 17:00, 7 March 2006 (UTC)
Anyone object
Rick, I did not claim that the book on Quantum Mechanics is the place to go for basic mathematical definitions. I merely claim that the reference proves that this is not original research on my part. It is true that N[X] is not a ring, but it is not to the point, for we don't need a ring. The extensions of numbers N→Z→Q match the extensions of polynomials N[X]→Z[X]→Q[X].
MFH, The first example given on Polynomial is in Z[X] but not in N[X]. The elements of N[X] are polynomials just as the elements of N are numbers.
Oleg, 'my' definition compares favorably to all the explanations in the article. If you or MFH know a rigorous (not rigurous) definition which is also comprehensable to the readers, how come it does not appear in the article ? Bo Jacoby 12:00, 8 March 2006 (UTC)
You are right and I am embarassed. Bo Jacoby 12:47, 8 March 2006 (UTC)
I wrote what I mean and I mean what I wrote: "the same solution". Equations having no solution do not have the same solution, even if they do have the same (empty) set of solutions. However, the relation as defined is symmetric and transitive, but not reflexive, so I must specify that R1 is the reflexive closure. Thank you. Bo Jacoby 14:45, 8 March 2006 (UTC)
Bo, all this argument is because we did not go to the bottom of things. Let me ask you:
Also, what is an indeterminate, X? You are avoiding this question, and you are trying to construct the integers using polynomials, which were not defined. You must construct polynomials first, please understand that. You can't just say let X be an indeterminate. Oleg Alexandrov ( talk) 15:59, 8 March 2006 (UTC)
Oleg - The set, N[X], of polynomials is defined recursively, just as the set, N, of natural numbers is defined recursively by Peanos axioms. You are not told what a natural number is, only that the axioms apply. Basicly the axioms says that if a is in N then so is a+1, and that 1 is in N. You are not told what 1 is, and you are not told what +1 means. From the axioms, addition and multiplication are defined, and the arithmetic rules, a+b=b+a. (a+b)+c=a+(b+c), a(b+c)=ab+ac, ab=ba, (ab)c=a(bc), a+b=a+c => b=c, and ab=ac => b=c are proved. N is closed under addition and multiplication: If a,b are in N then so is a+b and ab. N is characterized by these two conditions. (1): N is closed under addition and multiplication, and (2): 1 is in N. Now define N[X] by similar conditions: (1): N[X] is closed under addition and multiplication, (2): 1 is in N[X], and (3): X is in N[X]. You are not told what X is, only that the rules apply. That is a perfectly sound definition. A solution to an equation, f(X)=g(X), is a number, a, which substituted for X in the equation produces a true statement: f(a)=g(a). A solution is not a set of numbers, but a number. The equation X+2=0 has no solution in N. Nor has the equations X+3=0 a solution in N. So they do not have the same solution. The two equations do not have different solutions. They have no solutions.
Gandalf - No, I do not want X+2=1 (A) and X+3=2 (C) to be in the same equivalence class under the relation R1 , as the two equations do not have the same solution in N. But they are equivalent under the extended relation R2 because 2+2=1+3. So the two equations define the same integer, which is the solution to either in Z.
Rick - It is not nonsens what I wrote about long division. I explained the difference between exact and approximate solutions to polynomial equations, the point being that the root of a first degree polynomial like 3x-1 cannot be expressed exactly any more or any less than the roots of a fifth degree polynomial like x5-x-1. The expression x=1/3 says nothing more than 3x-1=0. I you want to know the root you must approximate: x=0.3333. This is done by long division. Your claim: "When the polynomial is of degree greater than four, it is not always possible to find exact expressions for the zeroes" is not true, because the polynomial itself is an exact expression for the roots. The descartes method is basicly useless and has but historical interest and does not belong in an introduction. Please hesitate reverting other peoples edit. Be open minded and don't reject new stuff until you understand it. The article is very bad by now. There is no proper definition, the history section is very narrow, there are no application, and a lot of the text is untrue. It need to be improved, but it cannot be improved when you just revert any change. I don't expect you to agree, but you should ask questions first and shoot afterwards. Bo Jacoby 01:54, 9 March 2006 (UTC)
You wrote:
Look, you are saying it is false that the equations have the same solution, and it is also false that they don't have the same solution. That can't be, logically. Oleg Alexandrov ( talk) 03:03, 9 March 2006 (UTC)
They have the same solution set (the empty set), which I think is really what matters here. Dysprosia 04:07, 9 March 2006 (UTC)
It is logical to say that "you and I are neither dating the same girl nor different girls" if at least one of us is not dating any girl. Now substitute "dating a girl" with "having a solution". A solution is not the same thing as the set of solutions - if the set of solutions is empty, then there is no solution. The relationship (X+a=b)R1(X+c=d) does not mean that the two equations have the same set of solutions, but the relationship is true EITHER if each of the two equations has a solution and these solutions are the same, OR if a=c and b=d. (Any equation must be equivalent to itself). This relationship implies that a+d=b+c, that is: (X+a=b)R2(X+c=d). This implication does not go backwards: The equivalence relation R2 is a proper extension to R1. It contains more relationships. Let Z=M/R2 be the new set of equivalence classes of equations. There is a one-to-one correspondance between N and the subset of Z of equivalence classes of equations having the same solutions in N . A number is identified with the equivalence class of equation having that number as a solution. This identification extends the concept of number from N to Z. Bo Jacoby 10:01, 9 March 2006 (UTC)
Thanks for your contributions so far. How do you explain b−a with a,b in N ? To me it is a solution to the equation X+a=b. (Postulating R2 without R1 would omit this explanation). The same approach explains fractions and radicals and algebraic numbers. What explanation is natural to you ? It is your turn to be constructive. I'll then resist the temptation of being too critical. :) Bo Jacoby 11:19, 9 March 2006 (UTC)
You are definining a set M={X+b=a | a,b in N}. You can't do that. You cannot define a set of equalities, an equality is not a set element. You should say the set of all pairs (X+b, a), now that's correct. Oleg Alexandrov ( talk) 16:02, 9 March 2006 (UTC)
Rick - Every child in the whole world has to learn what an integer is, so it must be well explained in wikipedia. By now it is not well explained. My approach obviously provokes not only a lot of constructiv mathematical discussion but also some impatient dogmatic opposition. So I challenge my honored opponents to write an explanation which is understandable to the child and acceptable to Oleg and even to myself. Do you think that is easy ?
Oleg - You agree to consider sets of numbers, sets of pairs of numbers, and even sets of pairs of polynomials, but you disagree to consider sets of equations. Why is that ? Isn't an equation a mathematical object too ? The math teacher writes on the blackboard: , points at it, and says: "This is an equation". An equation, like a polynomial, has several interpretations: unevaluated it is a formal mathematical object - fully evaluated it is a truth value. The equation X+a=b is unevaluated - it is not a claim that the two sides are equal polynomials. The polynomial "X+a" is also unevaluated - It is not a number, but either a function, x→(x+a), or an abstract mathematical object in its own right. The equation X+a=b may be considered a boolean function of an integer: x→(x+a=b). Here it even means (a,b)→(x→(x+a=b)) : to every pair, (a,b), there correspond a function that to a number x, tests whether x+a=b is true or not. If we stop talking about these equations and just talk about pairs then nobody will understand the meaning of the equivalence a+d=b+c. Your objection indicates that we should write fully: M = { (a,b)→(x→(x+a=b)) | a,b,x in N } rather than shorthand: M = { X+a=b | a,b in N } . Personally I prefer using leftarrow rather that rightarrow for specifying functions, f=(y←x) meaning y=f(x), because y is more important that x, but that is nonstandard. In that notation the definition would be like this: M = { ((x+a=b)←x)←(a,b) | a,b,x in N } Bo Jacoby 10:22, 10 March 2006 (UTC)
If you are trying to explain something to others, like above, you should use common notation, not your own notation. :) But I think I got it. Thanks.
Look, you are saying that the equations X+3=1 and X+4=2 define the same solution in N. That is wrong. Assume that the solution is the same, call it n≥0. By plugging it in the first equation you see that it does not satisfy the equation, contradiction. Oleg Alexandrov ( talk) 16:24, 10 March 2006 (UTC)
Oleg, I agree that the equations X+3=1 and X+4=2 do not have the same solution in N. Neither of the two equations have a solution in N. So the two equations are not equivalent under the relation R1 (which demands that two equations have the same solution in N). But they are equivalent under the extended equivalence relation R2, because 3+2=1+4. So they define the same element in Z. Remember that the Z-element b−a is the set of equations of the form X+c=d where a+d=b+c. Dysprosia's last remark is perfectly correct. Bo Jacoby 23:15, 13 March 2006 (UTC)
Oleg, going directly to R2, the approach cannot be reused for constructing fractions and algebraic numbers. I was not trying to get to both the intuitive motivation and the rigorous construction in one shot. I wrote the intuitive non-rigorous section
You then requested a rigorous presentation, claiming that it couldn't be made. I showed that it can be made, albeit confusing, as rigor often is.
Dysprosia, If you have a construction that is really standard, lucid, verifiable, and source-able, then write it down and let us use it. Let us not solve a solved problem. Bo Jacoby 13:42, 14 March 2006 (UTC)
I enjoyed the discussion. When your arguments were refuted, you did not change your mind regarding the conclusion, but you just invented new objections. So your argumentation is reverse, constructing arguments for your conclusion rather than deriving a conclusion from the arguments. Surely, that kind of argumentation can go on till we drop. My purpose is not to make you change your mind, because that probably can not be done anyway, but to improve the formulation. You did pinpoint weak spots in the text. By now the polynomial article is without motivation and application. That is bad. Also there is nowhere an explanation for the construction of integers, fractions and algebraic numbers in general. That is bad too. I trust that you guys will mend these defects, as you do not allow me to do it. Bo Jacoby 09:02, 15 March 2006 (UTC)
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)
Polynomials are important for a number of reasons: 1. A sum of polynomials is a polynomial 2. A product of polynomials is a polynomial 3. The derivative of a polynomial is a polynomial 4. The integral of a polynomial is a polynomial 5. Polynomials serve to approximate many functions, such as sine, cosine, and exponential.
This text from www.jsoftware.com contains motivation in a nutshell. There is presently no section on motivation in the article. Bo Jacoby 11:04, 22 June 2006 (UTC)
I'm not sure about the latest change in the lead from
to
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)
We don't get to decide what we would like the word "polynomial" to mean. The word has a well established meaning, and that meaning does not allow non-natural real number powers or division by a variable. Rick Norwood 13:24, 9 December 2006 (UTC)
The anons recent edit to the lead raises an interesting distinction between a purely formal definition of a polynomial as the sum of monomials, and equations which can be reduced to that form. For example Stewart Galois Theory defines polynomials as a formal sum. Hence the formal expression is not a polynomial but its expanded form is. Should we make this clear in the lead? -- Salix alba ( talk) 01:03, 5 November 2006 (UTC)
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)