Talk:Factorization

Factoring $$a^2 - 2 b^2$$
In my opinion, this section is a special case of $$a^2 - b^2$$ being  $$a^2 - 2 b^2$$, which yields  $$(a-\sqrt2 b)(a+\sqrt2b)$$ for one, and there is no way of doing this without radicals (assuming a and b are integers) like the editor asks for. Furthermore, it is stated as if the editor wanted help in his homework, but of course without knowing the editor, I will not pass judgement on his intentions, just that it sounds terrible in the article. In any case, unless the section is relevant, which does not seem so, it should be deleted, or at least until what is proposed is found.

--GabKBel (talk) 03:35, 5 July 2011 (UTC)

Old comments
Whoa! Language alert!
 * 15 factors into primes (verb)
 * x2 - 4 factorises (verb)
 * In mathematics, factorization (noun)
 * The aim of factoring (noun)

Is there a good reason why there are two flavours of each? -- Tarquin 21:25 Sep 22, 2002 (UTC)

Unfortunately, both factor and factorize are used as synonymous verbs, each being more common in a different context, and each having its own noun form. When discussing the problem of breaking down large numbers, "factorize" is almost always used. In all other contexts it's usually "factor". I prefer the latter because it's shorter, but use the former when talking about the problem for large integers.

There's also the difference between using an "s" or a "z". That's purely a British vs. American issue, so it would be fine to standardize one way or the other on a given page.

I think there is a fast method of factoring integer into primes, but it requires a quantum computer.

Table method (for quadratics)
A less used (hard to teach, easy to learn) method often involves creating a multiplication table.

For example, let's work with 6x2 - 17x + 12.

Multiply first and last terms. (72x2)

What multiplies into 72 (first term) and adds up to -17(x) (middle term)?

-9 and -8

In the table, place the first term in the first box and the last in the last box. Fit the (in this example) -9 and -8 in the remaining boxes. Find the GCF up and down and side to side for each row for the answer.

The answer would be (3x-4)(2x-3)

This method is very decisive and much faster than the others. However, there are a few special rules surrounding the method, but when all of the rules are followed, it works every time.


 * I am not sure this method is faster than the generalized AC method shown. In this example ac = 72. Examination of the factors of 72 gives -9 -8 = -17 = b, as above. From here we get:


 * 6x2 - 17x + 12 = (6x - 8)(6x - 9)/6 = (2)(3x - 4)(3)(2x - 3)/6 = (3x - 4)(2x - 3). —Preceding unsigned comment added by 150.176.192.118 (talk) 15:51, 11 January 2011 (UTC)

Sum/difference of two cubes
I'm here to learn, but shouldn't the factorization be (x-10)(x^2+10x+100) to expand to (x^3 - 1000)? Sparky 21:10, 15 April 2006 (UTC)
 * You're so right. My apologies! --Mets501talk 21:31, 15 April 2006 (UTC)
 * Haha no problem! Keep up the good work. Sparky 22:44, 15 April 2006 (UTC)

Also here is a problem: x^2-y^2+8y+4x-12 should I Factori "x^2-y^2" or "y^2+8y-12"first?

Factor it to
 * $$x^2+4x+4-(y^2-8y+16)=12+4-16$$
 * $$(x+2)^2-(y-4)^2=0$$

&mdash;M e ts501 talk 15:14, 28 May 2006 (UTC)


 * I am not sure where the equal sign comes from. This is also a difference of squares, so the final form is given as:
 * $$(x+2+y-4)(x+2-y+4)$$ which simplifies to $$(x+y-2)(x-y+6)$$ —Preceding unsigned comment added by 150.176.192.118 (talk) 16:00, 11 January 2011 (UTC)

FYI, I'm finding the description here very confusing. I've reread it several times. Is there a typo in the description of a^n - b^n and a^n + b^n - maybe left out an "even" in the wording? Anyway, the description is confusing.

Order of sections
I would have expected that the prime factorisation of integers would be before factorisation of polynomials, etc. JPD (talk) 16:56, 7 August 2006 (UTC)
 * Yes, you're right; that would be more logical. I've changed it. — Mets 501  (talk) 19:00, 7 August 2006 (UTC)

Difference of two cubes,forth powers,fifth powers,etc.
$$a^n-b^n=c^2-d^2$$ like $$7^3-4^3=48^2-45^2$$ Bhowmickr 07:15, 28 August 2006 (UTC)


 * Not very exciting, since any odd integer may be expressed as a difference $$c^2-d^2$$, with both c and d being integers. JoergenB 12:55, 1 November 2007 (UTC)
 * How does one express an integer of the form 4n+1 as a difference of squares of integers? that said, the difference in the original problem must either be 0 or -1 mod 4. —Preceding unsigned comment added by 150.176.192.118 (talk) 16:07, 11 January 2011 (UTC)
 * If a = 2n + 1, then a = (n + 1)2 − n2.—Emil J. 16:17, 11 January 2011 (UTC)

Ratio Method for factoring quadratics
A teacher at my schoool came up with this method. I like it.

You multiply (A)(C). Use B. Find two numbers that multiply to give AC, and add to give B. Let's say they are Z and Y

Now make two columns on paper...each with a ratio. A will be the first number in both ratios. Z will be the second number in the first ratio, and Y will be the second number in the second ratio.

Reduce if possible. Say you end up with A:Z and A:Y. the answer will be (a+z)(a+y).

This is what they teach in grade 10. and it's easier/more convenient than the methods described here.  jmatt1122   C V U (Talk)  20:49, 4 November 2006 (UTC)

Cleanup needed
The article starts in a nice manner: integers, polynomials and matrices are considered. Sections 1 & 2 still seem to preserve this.

Then, it goes in a quite wild way: Sections 3 - 6 are somehow "unclassified". Then, section 7 pops up with logics, and matrices are present only in the "see also" section.

Who has ideas to re-equilibrate this a little?

As a first suggestion, could one make a 1- or 2-line summary for each of sect. 3-6 into the corresponding chapter (sect.1 or 2), make sect.3 on matrices (or sect.3 : logics (since a little bit related to polynomial factoring), then sect.4: short summary on most important matrix decompositions).

I suggest to put details from sect.3-6 on (a) separate page(s).

Another idea would be a subsection of 2 with a title going somehow like "deterministic formulae and/or algorithms for speical cases".

(So one could know more easily what section would be useful for a given problem, e.g.)

Or, third idea, make some short sections at the beginning on "what the main body of this article not really is about: integer, matrix & logic factorization" with appropriate links, and then elaborate on the polynomial factoring methods (say, in increasing degree of the polynomials or some other order).

This is meant as a brainstorming - I'm missing better ideas, so please help. &mdash; MFH:Talk 22:08, 5 January 2007 (UTC)

Cubed polynomials plus 7 imaginary
This article needs information on how to factor cubed polynomials and also why is it 7 imaginary rather than plus or minus y?

Gradster1 23:11, 8 May 2007 (UTC)

Error in "other common formulas"
I found a mistake in the fourth formula, it would be good if someone could review the formulas that come after this.AV-2 04:49, 3 September 2007 (UTC)

BUT IT STILL DIDNT TELL ME WHAT THEY ARE!!!!!!!!!!! —Preceding unsigned comment added by 211.27.0.27 (talk) 05:54, 8 February 2009 (UTC)

Factoring by grouping
There is no reason that the factoring by grouping should use a trigonometry example. This unnecessarily excludes students unfamiliar with trigonometry from following the example. Brentt (talk) 21:55, 24 June 2008 (UTC)

Last link in the external links section
Am I the only one who thinks that this is of no value at all and that the entire purpose of that page is to promote their software? Anyone else thinks it should be removed? Almogo (talk) 15:18, 17 August 2009 (UTC)

I removed it. Not only was it pointed toward vendor software, the connection with factorization was very tenuous. 128.186.122.40 (talk) 13:33, 17 September 2009 (UTC)

Funny introduction to the article
In the introduction part of the article, the following is stated:

"  However, factors are not needed to divide evenly because they are still divisible by any number. For example, technically 3 and 8/3 are factors of 8. But, when factoring for tests teachers are looking for the even divisibility of the numbers.''  "

This seems to be written with certain math tests in mind, and I don't think it is relevant for the article as a whole. Any objections to deleting that part?

Trolle3000 (talk) 23:19, 3 October 2009 (UTC)


 * The sentence is misleading and completely misguided. In the ring of integers, 3 does not divide 8, period. In the ring of rationals, 3 and 8/3 are true factors of 8, not just "technically". What particular teachers are looking for in particular tests is irrelevant. — Emil J. 10:52, 5 October 2009 (UTC)

I think this page requires clean-up and/or attention
At least the talk page does; lots of old requests or concerns on this page seem to be unattended. Also I added the factor theorem. I was surprised to find it missing in the other section (along with some recently added content that doesn't seem to make any sense... --Xali (talk) 23:54, 17 October 2009 (UTC)

Discuss the usage difference between "factor" and "factorization"
The two are equivalent, though I would like to know where/when each term was popular. 18.111.42.150 (talk) 19:34, 15 August 2010 (UTC)
 * The two are not equivalent. For example, 7 · 11 · 13 is the prime factorization of 1001, whereas 7 (as well as 11 or 13) is a prime factor of 1001.—Emil J. 12:39, 16 August 2010 (UTC)

Does Factorization serve a practical purpose?
Does it have a use? Fxm12 (talk) 23:48, 14 November 2011 (UTC)


 * Yes, it does: the RSA is based on factorisation of large numbers is very slow. Hári Zalán (talk) 10:01, 6 March 2023 (UTC)

Keeps teachers employed. (Just kidding --- couldn't resist.) --Dagme (talk) 02:15, 1 May 2014 (UTC)

LaTeX Code?
Is there a way to export LaTeX Code from Wikipedia articles such as this? --Dagme (talk) 02:14, 1 May 2014 (UTC)

Refactoring and adding references
I have started a refactor of this page. The page, as originally envisioned (as I see it), was a gentle introduction to elementary factoring techniques. It was well done and was decently structured (but never completed). Over time, additions by various editors have disrupted the coherence of the article and introduced some subtle errors. In thinking about how to fix the page it became clear to me that the original organizing principle (doing a thorough job with quadratics in one variable and then move on to more complex polynomials) was too limiting for an encyclopedic article (it would be ok in a textbook). Thus, I am scrapping the original approach in reorganizing this page. I hope to keep the level of the article about the same and I will reuse all the quadratic stuff as examples. This will take a little time. Bill Cherowitzo (talk) 18:50, 3 November 2014 (UTC)

Assessment comment
Substituted at 02:05, 5 May 2016 (UTC)

Separating univariate polynomial factorization from bivariate polynomial factorization
Right now in the main (polynomials) part of the article, the sequence of subsections randomly alternates between univariate (U below), bivariate (B below), and trivariate (T below) polynomials as follows:
 * BBUUUBBBBBBTUB

I'm going to rearrange this to put all the univariate subsections first. Loraof (talk) 16:52, 15 September 2017 (UTC)
 * There is another important issue in the polynomial part of the article: although it refers to Polynomial factorization through a main template, its content differs dramatically from the content of the main article (normally, the content of a section with a template main should be a summary of the main article). Moreover, there is a undue weight given to methods that are used only in elementary teaching, and nowhere else; this is misleading, because a reader who do not read the article carefully may ignore that the theory of polynomial factorization is an active area of research, and that a large part part of this theory is interesting at elementary level (for example primpart-content factorization and square-free factorization).
 * Therefore, IMO, the article deserve more work than simply permuting sections. D.Lazard (talk) 17:35, 15 September 2017 (UTC)


 * Okay. The present article has almost nothing about factorization of anything but polynomials, and the substantial majority of it is about bivariate polynomials, while Polynomial factorization is exclusively about univariate polynomials.


 * Therefore, I propose to turn the present article, Factorization, into an article called Factorization of bivariate polynomials, and to move the present article's content on univariate polynomials over to Polynomial factorization. What do you think? Loraof (talk) 20:14, 15 September 2017 (UTC)
 * I have reread the article. I agree that some refactoring could be useful. However, the sections that you call "bivariate" are not really about the bivariate case; instead they refer either to the general multivariate case, with examples that are simply bivariate, such as the section about the factorization of monomials, or they present patterns referring to two subexpressions. In fact, the authors of the article use x, y, ... for indeterminates, and a, b, ... for subexpressions occurring in patterns (this would deserve to be explicitly said). Thus the classification could be between sections strictly devoted to univariate case and sections where the number of variables does not matter. For me, it is unclear, if this ordering is, or not, better than an order based on an increasing technicality.
 * About the move: you are wrong when saying that Polynomial factorization is only about the univariate case; it shows that the multivariate case reduces to univariate case through a recursion on the number of variables. Therefore, I suggest rather to move the article to Factorization (hand-written). D.Lazard (talk) 09:37, 17 September 2017 (UTC)


 * Okay, how about if I rename this article Factorization of polynomials by hand, remove the essentially contentless sections not about polynomials, and after the section on basic concepts put a section on univariate polynomials followed by a section on multivariate polynomials, mentioning that a and b can be indeterminates or polynomials? Loraof (talk) 19:19, 17 September 2017 (UTC)
 * Before moving this article, we should examine what doing with its title. In fact, "Factorization" is probably the first term which is searched when searching for either factorization of integers, or polynomials, or other kinds of objects. therefore, the tilt must be kept. Since all these subjects are strongly related, a dab page is not convenient. A WP:broad-concept article, would be possible; however the spirit of broad-concept article is being a kind of stub before writing a more developed article. This article looks like a broad concept article that has been developed for one of its aspects (polynomials) but not for the others (integers and other kinds of objects cited in the lead). I understand your suggestion as splitting the article into a broad-concept article, and a new article on polynomials. However, it would leave empty or almost empty the description of other factorizations cited in the lead.
 * On the other hand Integer factorization is only about algorithms, and I have not found any article describing the elementary methods (test of divisibility by 2, 3, 5, 11, and its use for trial divisions; Sieve of Eratosthenes, and its use for trial divisions, etc.).
 * For these reasons, I'll put clearer hatnotes to the sections, and expand section tags, with explanation of the desired content of such expansion. IMO, it is more important to expand these almost empty sections than moving the article.D.Lazard (talk) 13:28, 19 September 2017 (UTC)

Rewriting this article
I know, for a long time that this is article was a mess (see the above thread). A recent edit by showed me that it was full of mathematical errors an WP:OR opinions, mainly that a factorization must produce a simplification. Therefore I have rewritten the lead and the integer section, and started to rewrite the section on polynomials (not yet published). Doing that, I remarked another issue of this article: several hatnotes (there were too many) asserted that the article intends to be elementary. In fact, the simplest algebraic manipulations are detailed as if this were a first textbook for learning algebraic manipulations, and so many details are given that the main points become confusing. In the same time, the reader that is not supposed knowing simple algebraic manipulations is supposed to know what is a polynomial and what is the efficiency of a computation. Thinking about this, it appeared to me that the long section about factorization of polynomials was, in fact about factorization of expressions, as all methods apply when, for example, trigonometric functions are involved.

For these reasons, I have started to rewrite the whole article, keeping it at least as elementary as it was, but placing it in the right mathematical context, and, when it is relevant, giving some insight toward more sophistical concepts. I hope that the result will be convenient for beginners, as well as for mathematicians. D.Lazard (talk) 14:44, 21 March 2018 (UTC)
 * ✅. The announced rewrite is now complete. D.Lazard (talk) 17:58, 26 March 2018 (UTC)

Error
A small remark from a non-expert: I am quite certain the Fermat number 2**32+1 is 4294967297 and not 4295967297 as mentioned in the text. I think this is an error and should be corrected. [R.m.c. Ahn] — Preceding unsigned comment added by 86.82.164.5 (talk) 23:21, 12 September 2018 (UTC)
 * ✅. Good catch. --Bill Cherowitzo (talk) 03:55, 13 September 2018 (UTC)

Add Factorisation by inspection section to polynomial section
Request to add "factorisation by inspection" to factorising polynomials section. This method uses abstracted concepts from factor theorum method, deserves its own section. Raakkk (talk) 01:39, 19 May 2023 (UTC)


 * What size is it please 87.117.87.35 (talk) 18:31, 21 September 2023 (UTC)
 * I don't understand what size is ìt can you explain to the si 87.117.87.35 (talk) 18:33, 21 September 2023 (UTC)
 * A source is needed for including the method. I have never heard of a "factor theorum method". As far as I know, "factorization by inspection" cannot be very different from the content of section . If I am wrong, please, clarify your meaning of "factorization by inspection". D.Lazard (talk) 20:42, 21 September 2023 (UTC)