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Text and/or other creative content from this version of Polynomial was copied or moved into Properties of polynomial roots with this edit. The former page's history now serves to provide attribution for that content in the latter page, and it must not be deleted as long as the latter page exists. |
The contents of the Polynomial expression page were merged into Polynomial on Nov 6, 2014. For the contribution history and old versions of the redirected page, please see its history; for the discussion at that location, see its talk page. |
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This page has archives. Sections older than 365 days may be automatically archived by Lowercase sigmabot III when more than 10 sections are present. |
It seems to me that this article gets into advanced topics too quickly. They belong in the article, certainly, but perhaps in a later section, with the first part of the article covering the elementary properties of polynomials. What do you think? Rick Norwood ( talk) 16:06, 27 February 2019 (UTC)
The section that bothered me was "Notation and Terminology", which goes on at some length about the (important) distinction between a polynomial, a polynomial equation, and a polynomial function. I think this could be stated in language more accessable to a non-mathematician. Rick Norwood ( talk) 21:32, 27 February 2019 (UTC)
I have reverted the new section "Other bases" because;
Nevertheless a section on positional notation, could be useful for saying that this notation represents a polynomial in the basis with some constraints on the coefficients. It would also be useful to explain that the arithmetic operations in positional notations are the polynomial operations, except for the carries. Please, try a better version. D.Lazard ( talk) 20:32, 4 May 2019 (UTC)
I've long felt that our mathematics articles need some improvements to make them more accessible to the layperson without giving up any mathematical rigor. While I felt this I haven't done anything about it until now. I'm going to try with the very baby step. I think this article generally starts out great, but when I get to one section I thought it could be improved. I won't write out my entire rationale here, but it may be worth glancing at User:Sphilbrick/Mathematics articles if you don't think my proposal is an improvement. —Preceding undated comment added 20:59, 17 July 2020 (UTC)
I like this section. I would characterize it as very good but not excellent. It has one shortcoming that I think is easily resolved.
I like the progression of the left side of the page starting with the simplest polynomial (degree zero), including the special case of the x-axis and the more general case, followed by degree 1, 2 etc. Then showing the general case of degree n. I like the parallelism of having formulaic expressions on the left side, and nice looking graphs on the right side.
I have two concerns, one of which is almost trivial:
I'm trying to decide whether I should be troubled by having formulas on the left for degrees 0,1,2,3,n, while graphs for degrees 2,3,4,5,6,7. I've expressed the desire to add graphs for degrees 0 and 1, should I be troubled that we have a graph for degree for 4,5,6 and 7 but no formula? I'm not troubled but it may be worth discussing.
If others concur, I'll try reaching out to the editor who created the graphics. I think it would be easy to create similar graphics for degrees 0 and 1. if those editors aren't responsive I'll try the graphics lab.-- S Philbrick (Talk) 14:36, 18 July 2020 (UTC)
I'm happy to see the addition of graphs for the degree 0 and degree 1 polynomials. At the risk of being anal, we now have four different styles of graphs, obviously because the graphs were created by different people with somewhat different styles.
As a separate issue, the formulas chosen are obviously different from one another, but more so than necessary (unless this is deliberate and I'm missing the reason why). I think that it would be good practice to start with something basic:
f(x)=2
Then keep adding terms:
f(x)=3x+2
Then multiply by, say x-2:
f(x)= 3x^2 -4x -4
Possibly adding a scaling value if we want to keep the values within a containment region. Continue mutatis mutandes.
If this makes sense, looking to someone who can help us create the graphs.
I added two steps to the polynomial multiplication to show the intermediate steps.
I fully understand that people conversant with the subject will find the intermediate steps unnecessary, but many in our audience will find it helpful, and the more advanced we do can skip to the last step easily.
The rest of the section has some issues. It isn't that anything seems wrong, so it's hard to put my finger on it but it doesn't feel very organized. I'm in discussion with another editor about how it can be improved.-- S Philbrick (Talk) 14:27, 19 July 2020 (UTC)
The arithmetic section contains a statement:
Thus a sum of polynomials is always another polynomial.
That statement immediately follows an example. The word "thus" in mathematics is typically a synonym for "therefore" and typically means that the result follows from the immediately preceding statements. I trusted it is obvious that a single example cannot provide proof of a universal claim (obviously, a single example can prove the falsity of a claim but that's not what is at issue here.) Arguably, the word "thus" doesn't simply refer back to the immediately preceding example, but refers to the opening statement of the section:
Polynomials can be added using the associative law of addition (grouping all their terms together into a single sum), possibly followed by reordering (using the commutative law) and combining of like terms
If that's the intention, then it should be moved up but we have other things to discuss.
Arguably, we can claim that while the wording is a little sloppy, the polynomials constitute a set, and it happens to be true that the set is closed with respect to the operations of " addition, subtraction, multiplication, and non-negative integer exponents of variables." I think that's true (I confess I'm rusty), but such a statement requires a reference. There is a reference at the end of the sentence: Polynomials The reference suggests that pages one and two are relevant, which they are, but I reread them twice and don't see anything that makes the claim about the set being closed to those operations.
Additionally, if the set of polynomials is closed to the operations of " addition, subtraction, multiplication, and non-negative integer exponents of variables." we ought to make that statement, with a reference, not simply the statement that (closure exists) with respect to addition.
At a minimum, we need a reference supporting a claim, and separately we need to decide whether to make the claim narrowly about addition or more broadly. Plus, as noted we either need to remove the word "thus" if we want the statement to immediately follow an example, or we need to place it properly in the section.-- S Philbrick (Talk) 18:58, 20 July 2020 (UTC)
I'm trying to keep the terminology as simple as possible. It is understandable that some will feel that saying:
"a sum of polynomials is always another polynomial "
is simpler than saying:
"the set of polynomials is closed with respect to the operation addition"
However, the term " closed" is the mathematical way of making the statement, and there are many many references supporting the claim that polynomials are close with respect to addition (because that's the way it is said) but it's harder to find references to make the arguably simpler statement . For that reason, I introduce the notion of closure, which permitted the addition of a recent reference. — Preceding unsigned comment added by Sphilbrick ( talk • contribs) 18:06, 26 July 2020 (UTC)
In general, a sum of polynomials is always another polynomial
the addition in not the restriction of an operation defined on a larger set.
You can verify that the word "closed" does not appear in Operation (mathematics).
although it needs some word-smithing, and referencing. Making the assertion that the result of the operation produces another polynomial is a statement requiring a reference. It may seem obvious to the mathematically trained, but not to those who are not (our main audience) and it isn't trivially true; for example, the comparable statement about division is not true.the addition of polynomials is an operation that takes any two polynomials and produce always another polynomial, and is defined as follows.
When polynomials are added together, the result is another polynomial.
I had made some edits which were reverted twice. It was about changing the Summation Notation in Definition Section from:
to
The first one has a domain of all real numbers but 0 (with an indeterminate form 0^0 when x = 0), the second one has domain of all real numbers as is the case with polynomials. I understand that the former is simpler notation but since this is an important fundamental mathematical topic, I would argue that precision should be favoured over simplistic approach. -- Niteshb in ( talk) 01:17, 18 March 2021 (UTC)
I came to this page looking for a quick reference on how polynomials are defined in contemporary mathematics. The "Definition" section didn't really have this information, and furthermore was barely sourced. I decided I should help out and improve it, and spent about twelve hours putting together a much more thoroughly-sourced version of that section covering polynomials in both an elementary context and in mathematics in general. My work was reverted within a day with little ceremony. I would be perfectly happy if people wanted to keep working on it, but it doesn't seem sensible to me to throw out all my hard work in favor of something much terser that isn't as grounded in high-quality sources. Much of this article would really benefit from more thorough citations and I was doing my best to help.
The reasons cited for the reversion was that "a polynomial is not a function" and the change needs consensus. So, here I am seeking consensus. Considering the objection that "a polynomial is not a function," here's Serge Lang on the matter:
We now give a systematic account of the basic definitions of polynomials over a commutative ring …Consider an infinite cyclic group generated by an element . We let be the subset consisting of powers with . Then is a monoid. We define the set of polynomials to be the set of functions (emphasis mine) which are equal to except for a finite number of elements of . [1]: 23
Thomas W. Hungerford gives a very similar construction:
Theorem 5.1. Let be a ring and let denote the set of all sequences of elements of such that for all but a finite number of indices .
…
The ring of Theorem 5.1 is called the ring of polynomials over . [2]: 149
Of course, a sequence is a function with domain or , as he notes soon after:
...a polynomial in one indeterminate is by definition a particular kind of sequence, that is, a function (emphasis mine) . [2]: 151
Lest anyone wants to claim that defining polynomials as functions isn't done in a more beginner-friendly context, Lang also defines them as such in his book Basic Mathematics, a pre-calc textbook appropriate for high school students:
A function (emphasis mine) defined for all numbers is called a polynomial if there exists numbers such that for all numbers we have
- [3]: 318
Of course, this doesn't draw the same distinction between polynomials and polynomial functions that is made in more rigorous contexts, but that's why I described this style of definition separately.
It is true that other constructions are sometimes used, but I haven't seen any that are substantially different. For example, Birkhoff and Mac Lane in A Survey of Modern Algebra define a "polynomial form" as the form of an expression , where are elements in an integral domain and is an element of an integral domain of which is a subdomain. They then define a polynomial function as one with a definition that can be written in polynomial form. [4]: 61–63 Although I did make use of their book, I didn't go into this construction because it wasn't present in any of the other sources I was using, it's mainly a semantic distinction, and I didn't want to make the definition section too long. After all, Birkhoff and Mac Lane don't rule out the idea that an "expression in polynomial form" itself describes a function distinct from a polynomial function—given their definition, what else would it describe? Even so, I'm happy to include their definition explicitly for the sake of thoroughness if other people here feel that's most important.
Anyway, I think this makes an open-and-shut case for the worth of the edit I made. This is material from popular textbooks written by luminary mathematicians on this topic, so it belongs in this article. As such, I'd like to restore my edit. I'd also like to do other work on this article—there are entire sections with no citations, which needs to be fixed.
Mesocarp ( talk) 08:52, 5 September 2021 (UTC)
References
The content in articles in Wikipedia should be written as far as possible for the widest possible general audience.The number of students in grade 10 is several orders of magnitude larger than the number of working mathematicians, etc. The article at present contains a very nice discussion of polynomial arithmetic that is understandable by students in grade 10, and which is also understandable by working mathematicians (I co-wrote it, and I am one). Your edit placed a vastly more technical definition of these simple operations in an earlier section while leaving the elementary definition in place, lower in the article -- this breaks any principle of good writing and common sense.
The first sentence of WP:TECHNICAL isThe content in articles in Wikipedia should be written as far as possible for the widest possible general audience.
Your edit placed a vastly more technical definition of these simple operations in an earlier section while leaving the elementary definition in place, lower in the article -- this breaks any principle of good writing and common sense.
…it is not clear from your response whether you have understood that the function in the Hungerford definition is completely different from the function in your edit.
The definition of polynomial is more complex that needed, repeated twice (in introduction and not-so-formal definition) and not really well defined: "... polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables."
What is a coefficient and what is its role in the polynomial?. They are mentioned to exists and then not "used" in the definition. I think it is better to be pragmatical and start with an example, then try to make it clear what the operations allowed to variables and what is the meaning of "coefficients x (variable exponentiation)". What "a·x²" means in practice. "x²" is vector and "a" a transposed constact vector (sort of escalar product)? Not clear at all from the definition. Or maybe the coefficient are SU matrices and the result is of evaluation the polynomial is just another vector? Or maybe "a·x²" must just be interpreted as another new variable?
"Indeterminates" does not even appear in my English dictionary. "Variable" is well known and understood and is more "friendly" with terms used later in the classification of polynomial (univariate, multivariate, ...). "Indeterminates" distract attentions without providing any meaningful information. — Preceding unsigned comment added by 88.4.162.172 ( talk) 10:46, 31 December 2021 (UTC)
I would like to add the definitions of a special polynomial i.e. greatest common denominator(P,DP)=P and normal polynomial i.e. greatest common denominator(P,DP)=1. This is important in understanding the algorithms for symbolic integration. Any concerns? TMM53 ( talk) 09:17, 2 January 2023 (UTC)
This
level-4 vital article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
Text and/or other creative content from this version of Polynomial was copied or moved into Properties of polynomial roots with this edit. The former page's history now serves to provide attribution for that content in the latter page, and it must not be deleted as long as the latter page exists. |
The contents of the Polynomial expression page were merged into Polynomial on Nov 6, 2014. For the contribution history and old versions of the redirected page, please see its history; for the discussion at that location, see its talk page. |
Index
|
||||
This page has archives. Sections older than 365 days may be automatically archived by Lowercase sigmabot III when more than 10 sections are present. |
It seems to me that this article gets into advanced topics too quickly. They belong in the article, certainly, but perhaps in a later section, with the first part of the article covering the elementary properties of polynomials. What do you think? Rick Norwood ( talk) 16:06, 27 February 2019 (UTC)
The section that bothered me was "Notation and Terminology", which goes on at some length about the (important) distinction between a polynomial, a polynomial equation, and a polynomial function. I think this could be stated in language more accessable to a non-mathematician. Rick Norwood ( talk) 21:32, 27 February 2019 (UTC)
I have reverted the new section "Other bases" because;
Nevertheless a section on positional notation, could be useful for saying that this notation represents a polynomial in the basis with some constraints on the coefficients. It would also be useful to explain that the arithmetic operations in positional notations are the polynomial operations, except for the carries. Please, try a better version. D.Lazard ( talk) 20:32, 4 May 2019 (UTC)
I've long felt that our mathematics articles need some improvements to make them more accessible to the layperson without giving up any mathematical rigor. While I felt this I haven't done anything about it until now. I'm going to try with the very baby step. I think this article generally starts out great, but when I get to one section I thought it could be improved. I won't write out my entire rationale here, but it may be worth glancing at User:Sphilbrick/Mathematics articles if you don't think my proposal is an improvement. —Preceding undated comment added 20:59, 17 July 2020 (UTC)
I like this section. I would characterize it as very good but not excellent. It has one shortcoming that I think is easily resolved.
I like the progression of the left side of the page starting with the simplest polynomial (degree zero), including the special case of the x-axis and the more general case, followed by degree 1, 2 etc. Then showing the general case of degree n. I like the parallelism of having formulaic expressions on the left side, and nice looking graphs on the right side.
I have two concerns, one of which is almost trivial:
I'm trying to decide whether I should be troubled by having formulas on the left for degrees 0,1,2,3,n, while graphs for degrees 2,3,4,5,6,7. I've expressed the desire to add graphs for degrees 0 and 1, should I be troubled that we have a graph for degree for 4,5,6 and 7 but no formula? I'm not troubled but it may be worth discussing.
If others concur, I'll try reaching out to the editor who created the graphics. I think it would be easy to create similar graphics for degrees 0 and 1. if those editors aren't responsive I'll try the graphics lab.-- S Philbrick (Talk) 14:36, 18 July 2020 (UTC)
I'm happy to see the addition of graphs for the degree 0 and degree 1 polynomials. At the risk of being anal, we now have four different styles of graphs, obviously because the graphs were created by different people with somewhat different styles.
As a separate issue, the formulas chosen are obviously different from one another, but more so than necessary (unless this is deliberate and I'm missing the reason why). I think that it would be good practice to start with something basic:
f(x)=2
Then keep adding terms:
f(x)=3x+2
Then multiply by, say x-2:
f(x)= 3x^2 -4x -4
Possibly adding a scaling value if we want to keep the values within a containment region. Continue mutatis mutandes.
If this makes sense, looking to someone who can help us create the graphs.
I added two steps to the polynomial multiplication to show the intermediate steps.
I fully understand that people conversant with the subject will find the intermediate steps unnecessary, but many in our audience will find it helpful, and the more advanced we do can skip to the last step easily.
The rest of the section has some issues. It isn't that anything seems wrong, so it's hard to put my finger on it but it doesn't feel very organized. I'm in discussion with another editor about how it can be improved.-- S Philbrick (Talk) 14:27, 19 July 2020 (UTC)
The arithmetic section contains a statement:
Thus a sum of polynomials is always another polynomial.
That statement immediately follows an example. The word "thus" in mathematics is typically a synonym for "therefore" and typically means that the result follows from the immediately preceding statements. I trusted it is obvious that a single example cannot provide proof of a universal claim (obviously, a single example can prove the falsity of a claim but that's not what is at issue here.) Arguably, the word "thus" doesn't simply refer back to the immediately preceding example, but refers to the opening statement of the section:
Polynomials can be added using the associative law of addition (grouping all their terms together into a single sum), possibly followed by reordering (using the commutative law) and combining of like terms
If that's the intention, then it should be moved up but we have other things to discuss.
Arguably, we can claim that while the wording is a little sloppy, the polynomials constitute a set, and it happens to be true that the set is closed with respect to the operations of " addition, subtraction, multiplication, and non-negative integer exponents of variables." I think that's true (I confess I'm rusty), but such a statement requires a reference. There is a reference at the end of the sentence: Polynomials The reference suggests that pages one and two are relevant, which they are, but I reread them twice and don't see anything that makes the claim about the set being closed to those operations.
Additionally, if the set of polynomials is closed to the operations of " addition, subtraction, multiplication, and non-negative integer exponents of variables." we ought to make that statement, with a reference, not simply the statement that (closure exists) with respect to addition.
At a minimum, we need a reference supporting a claim, and separately we need to decide whether to make the claim narrowly about addition or more broadly. Plus, as noted we either need to remove the word "thus" if we want the statement to immediately follow an example, or we need to place it properly in the section.-- S Philbrick (Talk) 18:58, 20 July 2020 (UTC)
I'm trying to keep the terminology as simple as possible. It is understandable that some will feel that saying:
"a sum of polynomials is always another polynomial "
is simpler than saying:
"the set of polynomials is closed with respect to the operation addition"
However, the term " closed" is the mathematical way of making the statement, and there are many many references supporting the claim that polynomials are close with respect to addition (because that's the way it is said) but it's harder to find references to make the arguably simpler statement . For that reason, I introduce the notion of closure, which permitted the addition of a recent reference. — Preceding unsigned comment added by Sphilbrick ( talk • contribs) 18:06, 26 July 2020 (UTC)
In general, a sum of polynomials is always another polynomial
the addition in not the restriction of an operation defined on a larger set.
You can verify that the word "closed" does not appear in Operation (mathematics).
although it needs some word-smithing, and referencing. Making the assertion that the result of the operation produces another polynomial is a statement requiring a reference. It may seem obvious to the mathematically trained, but not to those who are not (our main audience) and it isn't trivially true; for example, the comparable statement about division is not true.the addition of polynomials is an operation that takes any two polynomials and produce always another polynomial, and is defined as follows.
When polynomials are added together, the result is another polynomial.
I had made some edits which were reverted twice. It was about changing the Summation Notation in Definition Section from:
to
The first one has a domain of all real numbers but 0 (with an indeterminate form 0^0 when x = 0), the second one has domain of all real numbers as is the case with polynomials. I understand that the former is simpler notation but since this is an important fundamental mathematical topic, I would argue that precision should be favoured over simplistic approach. -- Niteshb in ( talk) 01:17, 18 March 2021 (UTC)
I came to this page looking for a quick reference on how polynomials are defined in contemporary mathematics. The "Definition" section didn't really have this information, and furthermore was barely sourced. I decided I should help out and improve it, and spent about twelve hours putting together a much more thoroughly-sourced version of that section covering polynomials in both an elementary context and in mathematics in general. My work was reverted within a day with little ceremony. I would be perfectly happy if people wanted to keep working on it, but it doesn't seem sensible to me to throw out all my hard work in favor of something much terser that isn't as grounded in high-quality sources. Much of this article would really benefit from more thorough citations and I was doing my best to help.
The reasons cited for the reversion was that "a polynomial is not a function" and the change needs consensus. So, here I am seeking consensus. Considering the objection that "a polynomial is not a function," here's Serge Lang on the matter:
We now give a systematic account of the basic definitions of polynomials over a commutative ring …Consider an infinite cyclic group generated by an element . We let be the subset consisting of powers with . Then is a monoid. We define the set of polynomials to be the set of functions (emphasis mine) which are equal to except for a finite number of elements of . [1]: 23
Thomas W. Hungerford gives a very similar construction:
Theorem 5.1. Let be a ring and let denote the set of all sequences of elements of such that for all but a finite number of indices .
…
The ring of Theorem 5.1 is called the ring of polynomials over . [2]: 149
Of course, a sequence is a function with domain or , as he notes soon after:
...a polynomial in one indeterminate is by definition a particular kind of sequence, that is, a function (emphasis mine) . [2]: 151
Lest anyone wants to claim that defining polynomials as functions isn't done in a more beginner-friendly context, Lang also defines them as such in his book Basic Mathematics, a pre-calc textbook appropriate for high school students:
A function (emphasis mine) defined for all numbers is called a polynomial if there exists numbers such that for all numbers we have
- [3]: 318
Of course, this doesn't draw the same distinction between polynomials and polynomial functions that is made in more rigorous contexts, but that's why I described this style of definition separately.
It is true that other constructions are sometimes used, but I haven't seen any that are substantially different. For example, Birkhoff and Mac Lane in A Survey of Modern Algebra define a "polynomial form" as the form of an expression , where are elements in an integral domain and is an element of an integral domain of which is a subdomain. They then define a polynomial function as one with a definition that can be written in polynomial form. [4]: 61–63 Although I did make use of their book, I didn't go into this construction because it wasn't present in any of the other sources I was using, it's mainly a semantic distinction, and I didn't want to make the definition section too long. After all, Birkhoff and Mac Lane don't rule out the idea that an "expression in polynomial form" itself describes a function distinct from a polynomial function—given their definition, what else would it describe? Even so, I'm happy to include their definition explicitly for the sake of thoroughness if other people here feel that's most important.
Anyway, I think this makes an open-and-shut case for the worth of the edit I made. This is material from popular textbooks written by luminary mathematicians on this topic, so it belongs in this article. As such, I'd like to restore my edit. I'd also like to do other work on this article—there are entire sections with no citations, which needs to be fixed.
Mesocarp ( talk) 08:52, 5 September 2021 (UTC)
References
The content in articles in Wikipedia should be written as far as possible for the widest possible general audience.The number of students in grade 10 is several orders of magnitude larger than the number of working mathematicians, etc. The article at present contains a very nice discussion of polynomial arithmetic that is understandable by students in grade 10, and which is also understandable by working mathematicians (I co-wrote it, and I am one). Your edit placed a vastly more technical definition of these simple operations in an earlier section while leaving the elementary definition in place, lower in the article -- this breaks any principle of good writing and common sense.
The first sentence of WP:TECHNICAL isThe content in articles in Wikipedia should be written as far as possible for the widest possible general audience.
Your edit placed a vastly more technical definition of these simple operations in an earlier section while leaving the elementary definition in place, lower in the article -- this breaks any principle of good writing and common sense.
…it is not clear from your response whether you have understood that the function in the Hungerford definition is completely different from the function in your edit.
The definition of polynomial is more complex that needed, repeated twice (in introduction and not-so-formal definition) and not really well defined: "... polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables."
What is a coefficient and what is its role in the polynomial?. They are mentioned to exists and then not "used" in the definition. I think it is better to be pragmatical and start with an example, then try to make it clear what the operations allowed to variables and what is the meaning of "coefficients x (variable exponentiation)". What "a·x²" means in practice. "x²" is vector and "a" a transposed constact vector (sort of escalar product)? Not clear at all from the definition. Or maybe the coefficient are SU matrices and the result is of evaluation the polynomial is just another vector? Or maybe "a·x²" must just be interpreted as another new variable?
"Indeterminates" does not even appear in my English dictionary. "Variable" is well known and understood and is more "friendly" with terms used later in the classification of polynomial (univariate, multivariate, ...). "Indeterminates" distract attentions without providing any meaningful information. — Preceding unsigned comment added by 88.4.162.172 ( talk) 10:46, 31 December 2021 (UTC)
I would like to add the definitions of a special polynomial i.e. greatest common denominator(P,DP)=P and normal polynomial i.e. greatest common denominator(P,DP)=1. This is important in understanding the algorithms for symbolic integration. Any concerns? TMM53 ( talk) 09:17, 2 January 2023 (UTC)