By "polynomial factorization over the integers", I mean problems and solutions like the following:


Find a factorization into irreducible polynomials for

  • $24x^2 +x - 10$ and
  • $5x^3 -2x^2-x+24$.


  • Suppose a factorization $24x^2+x-10=(ux+v)(u'x+v')$ where $0<u\leq u'$, $uu'=24$, and $vv'=-10$, and look for tuple $(u,v,u',v')$ satisfying $uv'+vu'=1$. By evaluating $uv'+vu'$ at all 32 guesses we find $24x^2+x-10 = \boxed{(3x + 2)(8x-5)}$.
  • $5x^3 -2x^2-x+24$ has a linear factor if and only if they have a rational root whose numerator divides the $24$ and whose denominator divides the $5$. By evaluating the polynomial at all 32 guesses, we find that $-\tfrac{8}{5}$ is a root. Through polynomial long division, the quotient of the original polynomial by $5x+8$ is found to be $x^2-2x+3$. Since the latter is irreducible, we have $5x^3 -2x^2-x+24=\boxed{(5x + 8)(x^2 - 2x + 3)}$.

This schema is set apart from other polynomial problems and techniques in that it

  • fundamentally relies on integer factorization (of the constant term and leading coefficient),
  • requires making a potentially large number of guesses (depending on the number of factors found for the coefficients), and
  • tends to produce reducible polynomials and linear factors at a frequency much higher than if random integer coefficients were chosen (because of the choices made by the authors of such problems).

That background aside, I'm asking two questions: one historical, one normative.

  1. Who introduced polynomial factorization over the integers into secondary school curricula, and what was their justification for doing so? I've had trouble finding historical documents and secondary sources that would answer this question—any references pointing in the right direction would be greatly appreciated.
  2. Why should secondary-school students be taught about polynomial factorization over the integers? I'm looking for evidence—personal experience, studies in mathematical pedagogy, applications in science/technology/engineering—that teaching this topic furthers some goal of the educational system.
  • 1
    $\begingroup$ All mathematical subjects taught in schools tend to accumulate a crust of useless garbage and antiquated customs. Examples: long division, factoring polynomials, rationalizing the denominator in expressions like $1/\sqrt{2}$, and most of the techniques of integration taught in 2nd-semester calculus. At the K-12 level, the most common reason for not eliminating this crust is that K-12 is mainly a form of daycare and a way of keeping kids from entering the labor market and driving down wages, so we need to keep them busy. Assiduousness in factoring polynomials is also a form of virtue signaling. $\endgroup$
    – user507
    Feb 28 '18 at 6:43
  • $\begingroup$ @BenCrowell: Might agree with the first sentence, but disagree with the examples. Long division is required to understand why rationals have repeating decimals, polynomial division, the Number of Roots theorem, etc. Pretty good justification for factoring polynomials is given in answers below, I think. $\endgroup$ Feb 28 '18 at 18:46

"any references pointing in the right direction would be greatly appreciated"

Reference [1] below is probably where you want to look. A few years ago I tried to obtain a copy of [1], but I was not able to find one. However, I do not have access to interlibrary loan. If you have access to a university’s interlibrary loan, then you might be able to get a copy --- perhaps from here, for example. Just now I found a copy of [1] at this web page, but I don’t know anything about the legitimacy of this source (legal and/or internet-safety).

A few years ago I looked through a copy of [2] that a nearby university library has (some notes I made on this book can be found on pp. 2-6 here), but I don’t remember to what extent [2] includes anything that you are specifically asking about. Nonetheless, I think it is likely that [2] would be in the bibliography of anything you’re interested in (that is reasonably thorough), so I’ve included [2] for literature search purposes.

Because neither of these references is very well known, I’ve included the most complete bibliographic information I have for them.

[1] Hobart Franklin Heller (1901-1981), Concerning the Evolution of the Topic of Factoring in Textbooks of Elementary Algebra Published in England and the United States from 1631 to 1890, Keystone Publishing Company (Berwick, Pennsylvania), 1940, vi + 165 pages.

This is the published version of Heller's 1940 Ph.D. dissertation (under William David Reeve, 1883-1961) at Columbia University.

[2] Amy Olive Châteauneuf (1901?-1988?), Changes in the Content of Elementary Algebra Since the Beginning of the High School Movement as Revealed by the Textbooks of the Period, Ph.D. Dissertation (under John Harrison Minnick, 1877-1966), University of Pennsylvania, 1929, x + 191 pages.

This also exists as a book published by Westbrook Publishing Company (Philadelphia, Pennsylvania) in 1929 (x + 191 pages). Reviewed by: Lao Genevra Simons (1870-1949) in Mathematics Teacher 24 #1 (January 1931), pp. 58-59; Ernst Rudolph Breslich (1874-1966), The School Review 39 #3 (March 1931), pp. 229-230.

  • $\begingroup$ Since Heller's Ph.D. was done at Columbia, you could go to the source: clio.columbia.edu/catalog/1936041 The hardcopy appears to be stored offsite and considered "rare". There's a link to Google Books, but Google doesn't have an e-copy either. $\endgroup$
    – shoover
    Feb 28 '18 at 16:47
  • $\begingroup$ Silly me. WorldCat: worldcat.org/oclc/559331230 . Find a library near you, since it's almost certainly non-circulating. $\endgroup$
    – shoover
    Feb 28 '18 at 16:58

If we can do something with integers, we can do it with polynomials too

Things like adding, subtracting, multiplying, dividing, factoring. At least, that's how I framed these kinds of topics when I taught remedial algebra classes that focused heavily on algebraic manipulations (polynomials, rational expressions, radical expressions). I'm by no means an expert in algebra or number theory (nor do I expect more than a tiny faction of students to study these things), but in Knapp's (basic) algebra book, he says one unifying theme is "the analogy between integers and polynomials in one indeterminate over a field." In some sense, polynomials are just "nice" and our mathematics has evolved around this fact.

We have a very limited toolset for solving equations analytically

We can use our trusty arithmetic operations to solve linear equations.

For nonlinear things, we're in luck if we happen to have access to the inverse $f^{-1}$ of some function $f$, and our equation happens to be linear in $f(x)$ or, more generally, linear in "$f$ of something linear" (still more generally, when our equation arises from the composition of any number of functions with known inverses). For example, $3e^{x - 2} + 1 = 5$ is linear in $e^{x - 2}$, and we can invert exponential functions.

We have exact formulas for 2nd, 3rd, or 4th degree polynomials, essentially based on on clever rewriting ("completing the square", for example, or "depressing the cubic") and the inverse idea above. Unfortunately, we don't get formulas for 5th degree polynomials and beyond.

However, we're also in luck if we can factor a polynomial, precisely because it converts a nonlinear equation into linear equations. We can find solutions to $24x^2 - x + 10 = 0$ because we know $24x^2 - x + 10 =(3x+2)(8x−5)$, and that a product is $0$ when at least one of its factors is $0$; we just need to solve $3x + 2 = 0$ and $8x - 5 = 0$.

Students learn that "factoring leads to solving" for polynomials, and it helps with (nice) expressions that aren't polynomials. Just today I asked a calc class to find all $x$-values for which the curve $y = e^x \sec(x)$ has horizontal tangents, which leads to solving $e^x\sec(x) + e^x \sec(x) \tan(x) = 0$. Nobody puts forth any ideas for solving the latter equation, until I ask how they normally solve polynomial equations. Now that "factoring" is in the air, we factor, and get our solutions.

To summarize this second theme, factoring is basically one of the only two tools we have for solving equations with one indeterminate (I can't think of a third). Because of this, a large percentage of equations that a student will be expected to solve analytically rely on factoring to some extent. If nothing else, factoring polynomials over the integers is a good introduction to factoring in general, and this way of thinking.


Here's a response somewhere between the historical and normative questions. Recall that most math courses have an idiom of definitions/theorems/proofs to be presented, and then support by way of homework exercises. Arguably, we sometimes lose sight that the exercises are really just verifying that the actual mathematical content (definitions/theorems) have been learned. In some cases, it is a challenge to extract exercises relevant to the concepts being presented. Also, math courses lose coherency and become more garbled the more that proofs are taken out of the course.

In that light, factoring polynomials in integers is a best-we-can-do response to giving intuition and examples relevant to topics such as the Remainder Theorem, the Factor Theorem, and the Fundamental Theorem of Algebra. It also gives examples of a crucial step in Completing the Square (example in Al-Khwarizmi's Al-jabr), used to prove the Quadratic Formula.

It's true that in practice the techniques may seem very limited, and that is so. But we might also ask: How else can we give any examples of solutions to equations other than the forms $ax + b = 0$ or $ax^2 + b = 0$? How else can we build intuition regarding the machinery needed to prove the preceding theorems?

Now, if a course is taught on the "faith-based math" idiom that is more widespread nowadays, then the proofs of the Remainder/Number of Roots theorems may fade into the background, and the factoring machinery that they use may not seem as essential. But ultimately the Fundamental Theorem of Algebra (etc.) is intimately tied to the idea of factoring, and it's hard to think of how that understanding would be otherwise scaffolded.

  1. Historical development: No clue. Dave's answer studly as usual.

  2. Rationale: Based on personal experience and intuition (which seems to even be allowed this time!)

a. This fits into the "why should we learn this" now approaching a category of questions, deserving a tag. It's a legitimate question of course, but non-trivial. It's not as if everything has had some sort of robust in-use double-blind statistical proof of impact. Why do we learn Shakespeare?

Note, I am NOT dismissing the question--it's still relevant even for Shakespeare. But it's non-trivial. Perhaps the real answer is that it was just "something we could do", therefore we did it, even though it appealed to teachers and writers, but did not benefit students and wasted their time. But also perhaps we could make at least a thought experiment of eliminating this topic (or other topics in math or Shakespeare) and then find out years later that there was a damage from the students not learning the topic and the cost/benefit of time versus benefit would pay for itself.

b. Sometimes learning a tool does more to build muscles than perhaps to be the tool itself.

For instance we force doctors to take calculus, sometimes with tricky integration methods like integration by parts. They don't really generally need it but it does force them to learn some algebra and comfort with mathematical topics in general. That maybe pays off when they have some feel for ratios (amount of drug per pound of patient), etc. even though it is a simpler topic. Or tangentially to reading papers where some statistics is discussed (note, I'm not even saying calculus measure theory relation to stats, but just having to deal with some Greek looking math goop on the page for once and feeling more confident when dealing with some in the paper. And one can easily make the argument that dealing with stats itself might be more efficient. But still it is a collateral benefit.)

Learning to factor and play with equations in MATH is probably building some general muscles that will be helpful when there are manipulations involved in partial fractions or integration by parts. (And yes, you can make the argument we could just build the muscles when we do these topics...and maybe that is right or maybe some muscle building ahead of time makes that stuff go down easier.)

c. Also, maybe this is minor but in the sciences (and business) it is common to rearrange and equation and get some insights out of it. For instance, the quadratic function allows you to calculate the total (given x), but separating into factors might have some physical relationship involved in the multiplication (like number of wells times average production giving total production...or whatever.) So, it's not uncommon that moving some terms around in a physics, engineering or even business analysis problem gives you an insight. So a little factoring muscle might be useful.


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