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Eisenstein’s criterion:

If you have a polynomial with integer coefficients (and a non-zero constant term) and there’s some prime number $p$ such that

  • $p$ goes into every coefficient but not the first (or last) and
  • $p^2$ doesn't go into the last (or first, respectively)

then your polynomial doesn’t factor into two polynomials with integer coefficients.

(But the converse isn’t true. That is, if the criterion fails, the polynomial might nonetheless not factor.)

This is true of polynomials of any degree, but is most useful for quadratics, for two reasons:

  1. Those are the polynomials that are most frequently factored over the integers in high school.
  2. It’s easy to see why the criterion holds true for quadratics.[1]

Its use is as a quick check for non-factoring, so as not to waste time trying to factor. Thus, for example, a student can tell at a glance that $x^2+9x+24$ doesn't factor (because $3|9$ and $3^2 \nmid 24$), without checking which of the many factors of $24$ might sum to $9$. And can tell that $x^2-10x+60$ doesn't factor without checking factors of $60$.

However, this criterion doesn’t seem to be taught in high school at all, which makes me wonder whether doing so is pedagogically unsound. So my question is: Should one teach this in a high-school algebra class? For quadratics at least? Why or why not?


[1] It’s especially easy to see for $x^2+bx+c$. There, that $p$ goes into the $b$ means that the $r,s$ in $(x+r)(x+s)$ must both or neither be divisible by $p$; but then $c=rs$ must be divisible by $p^2$ or not by $p$ at all.

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"If you have a polynomial with integer coefficients" Doesn't all polynomials per definition have integer coefficients? –  Quora Feans Mar 30 at 16:30
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@Quora Feans You can always define polynomials over a ring R (R[x]) and factorability depends on the ring properties (algebraically closed field, etc). –  Chris C Mar 30 at 16:49
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@QuoraFeans even high-school algebra deals with polynomials with rational coefficients, such as $\frac12x^2+2x+2$. –  msh210 Mar 30 at 17:54

6 Answers 6

One drawback I see is that it is only marginally better than the standard algorithm of factoring $c$ and seeing which factors add up to $b$.

To apply Eisenstein's, you must find a factor of b or c and then test it on the other factor. So factoring occurs in either situation.

While Eisenstein's is a little bit faster, it doesn't lead to a solution in the case it does factor.

I see Eisenstein's as more useful in the cubic case, and I think it would be useful to introduce for the quadratic if only to generalize to higher degrees.

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1  
Thanks, but note that the criterion only says that a polynomial doesn't factor. Thus, its use is as a quick check for non-factoring, so as not to waste time trying to factor. (Perhaps I should make that clearer in the question.) So it'd be used to quickly check a couple of low prime numbers that, at a glance, go into the coefficients. –  msh210 Mar 30 at 13:59
    
@msh210 That's not a bad idea. –  Brian Rushton Mar 30 at 14:17

No, one should not teach Eisenstein's criterion in a high school class, because that puts far too much focus on factoring. In my opinion, a high-school algebra class ought to give students mathematical skills that are useful in non-mathematical contexts, and this would detract from that goal.

As an alternative, one can ask questions like: How does the graph of $x^2-10x+24$ compare with the graphs of $x^2-10x+23$, $x^2-10x+25$, and $x^2-10x+26$? What are the numerical values of the roots, and how do they differ for the different polynomials?

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I agree. Though factoring is helpful for getting students to understand algebra better, it's not really an end unto itself, and there's little point in introducing special tricks to make factoring of quadratics slightly easier. –  Jim Belk Mar 30 at 15:36
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(1) Dependence of graphs on parameters is a considerably more sophisticated issue than the "arithmetic" of factoring. (2) Such "arithmetic" has many practical applications (crypto, coding) as well as reappearing in more sophisticated mathematics. –  paul garrett Mar 30 at 17:22
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I somewhat object to your first paragraph, which is the justification for your answer of No, and point to the classic argument of saying "high-school English classes ought to give students the writing skills that are useful in non-academic (real-world, etc) contexts, and studying Shakespeare would detract from that goal." –  Benjamin Dickman Mar 30 at 20:41
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@BenjaminDickman, you understand me well. I think it makes sense for high school students to study a bit of Shakespeare: a couple sonnets, a scene or two, maybe a film adaptation, enough to get the cultural reference. But asking them to study the text of a whole five acts requires more struggle with archaic language and monarchical concerns than I think worthwhile. There's plenty of literature to teach from the past century. –  Matt F. Mar 31 at 11:08
    
The idea that the goal of education is to make one better prepared for "the real world" (whatever that is) is simply ludicrous. Think for yourself about how often you use anything you ever learned in school (outside of the occasions when you are teaching others those exact concepts). The goal of an education is not to teach practical things, but to exercise and further develop one's ability to think. –  Shay Apr 4 at 4:37

I maintain that what you want to focus on is sense making (MESE 1) (MESE 2). If you can present Eisenstein's criterion as a way of tackling such problems that helps to promote students' critical thinking about algebra, then I would say go for it. (I'd also be interested to look at your curricular materials!)

To illustrate my more general feeling, I look at your example: Is $x^2 + 9x + 24$ reducible?

Even with Eisenstein's criterion, one must still find a prime $p$; of course, here the choice $p=3$ is pretty immediate, so you can, indeed, employ this method of solution to show that the expression is irreducible. More generally, though, there can be a question of which prime to use...

From my vantage point, I'd rather a student remark: $b^2 - 4ac = 9^2 - 4(1)(24) = 81 - 96 < 0$, so there are no real solutions. (In particular, the polynomial is irreducible [over e.g. $\mathbb{Z[x]}$].)

Alternatively, I wouldn't mind seeing a solution like: $x^2 + 9x + 24 = (x+\frac{9}{2})^2 + $ something positive; so there are no real solutions (hence no integer solutions) and the polynomial cannot be factored in $\mathbb{Z}[x]$.

(To be clear: I would not use the $\mathbb{Z}[x]$ notation in a high school class.)

As a third method, a student might note that a hypothetical factored expression would have to look like $(x-a)(x-b)$ for $a, b > 0$; you need $ab = 24$ and $a + b = 9$, but the latter constraint means the maximum of $ab$ occurs when $a = b = 9/2$; observe $(9/2)^2 = 81/4 < 96/4 = 24$. Therefore, no such $a, b$ exist; so the expression cannot be factored.

The three methods suggested here are similar on a deep level, but how related would your high school students find them? What other methods could be used? There is already a lot that students can explore without adding on an additional criterion (at least one that I did not see until a course on Group Theory).

Moreover, I would especially like it in any of the suggested alternatives if the student could give an explanation of why these methods are viable, i.e., at a level that makes them understandable to other students in the course who would not think of such approaches.

Summary: Incorporating Eisenstein's criterion into a high school course on algebra would be nonstandard but not (for any reasons I see) obviously worth objecting to; the question remains whether or not you can make this criterion meaningful, and do so in such a way that students who understand the approach can give their own explanations - ones that are understandable to others in the class.

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I see little sense in introducing Eisenstein's criterion (a pretty big cannon) only to have a slight advantage when trying to factorize quadratics. It's really not justified. The strength of Eisenstein's criterion is in proving large polynomials irreducible. This certainly can be taught at the high school level since it's not that complicated and quite intuitive. But then the students need to be asked for the irreducibility of such polynomials as $x^{100}+2$ or $x^{100}-1$. It is then a very nice example of how to solve questions where a naive direct approach would lead to disaster.

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While this is a reasonable viewpoint, in fact I find the traditional "algebra" curriculum in middle school and high school quite impoverished of any genuine algebra, and Eisenstein's criterion is wonderfully easy, yet moves in the direction of real things. A few cyclotomic polynomials? Kummer polynomials $x^n-a$? Those things appealed to me at the time... and are defensible by their applicability to signal processing coding, and to crypto, not to mention "higher algebra", etc. –  paul garrett Mar 30 at 21:47
    
yes, these applications are certainly accessible. –  Ittay Weiss Mar 30 at 21:52

Firstly for quadratic equations, the quadratic formula will tell all. Is b^2-4ac a perfect square ? I think that is easier then checking for common primes. Secondly you didn't state the criteria right. p may not divide the coefficient of the highest term, however, p but not P^2 may divide the constant term. The point of Eisenstein's criteria is to give a criteria for factorization in cases when finding the roots of the polynomial isn't reasonable (think degree >4). I wouldn't think about Eisenstein's criteria unless faced with a polynomial that I couldn't factor. That's my two cents.

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Good point about quadratics. I think I stated the criterion (or the criterion with a corollary) correctly. –  msh210 Mar 31 at 2:37
    
This includes polynomials with rational solutions $\frac{1}{2}x^2+\frac{3}{4}x+\frac{1}{2}$ These some students might think don't factor even if it has factorization $\left(\frac{1}{2}x+\frac{1}{2}\right)\left(x+\frac{1}{2}\right)$ –  ruler501 Mar 31 at 3:28
    
@msh Suppose p=2, wouldn't x^2 +2x +1 satisfy the Eisenstein criteria you state ? More generally, as I read your Eisenstein criteria, (x+1)^p would qualify. Am I wrong ? –  aginensky Apr 1 at 16:08
    
To me, Eisenstein's criteria deals with very subtle issues about arithmetic and factorization that are not appropriate for most hs students. Again just imho. –  aginensky Apr 1 at 16:09
    
@aginensky, no, $2$ doesn't go into all but one of the terms of $x^2+2x+1$ (nor does $p$ go into all but one of the terms of $(x+1)^p$). –  msh210 Apr 1 at 17:20

I think it is perfectly fine to introduce as early as possible. It is a useful criterion, insofar as it can be executed quickly without holding much information in one's mind.

Probably the main reason it is not part of the traditional high school curriculum is just inertia, both in the textbooks and in the education of high school math teachers.

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