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This seems like such a niche trick to teach students when factoring polynomials. Like, the polynomials I've seen textbooks ask students to factor by grouping seem so cherry picked that I can't imagine this is a meaningful thing to show students.

  • For a quadratic $ax^2+bx+c$ you have a see a clever way to partition that $b$ as $d+e$ such that $ax^2+dx$ and $ex+c$ have a common linear factor. But what's the point of asking a student to play this guess-and-check game over the guess-and-check game for $r$ and $s$ in $a(x-r)(x-s)$?
  • For a cubic to be factorable by grouping I think it must be of the form $a(x-b)(x^2-c)$, where the quadratic factor can't have more than two summands.

So what's the motivation for considering this as a topic in the curriculum?

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    $\begingroup$ How else would you factor anything, for example, $x^2 + 6x + 8$, other than factoring by grouping? $\endgroup$
    – Chris Cunningham
    Nov 5, 2020 at 18:30
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    $\begingroup$ I think you're omitting one of the main points, which is that much of this work is useful for reinforcing other skills and concepts. Many beginning algebra students (indeed, even algebra 2 students and precalculus students) would have difficulty in recognizing that $(2x + 3y - 2)$ is a common factor of $(4x + 6y - 4)(x + y + 2) - 5(x + y - 2)(2x + 3y - 2),$ which of course is a rather artificial example, but one can easily come up with non-artificial examples involving exponential and/or trigonometric functions (say, when factoring to apply the first derivative test). $\endgroup$
    – Dave L Renfro
    Nov 5, 2020 at 21:52
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    $\begingroup$ Nonetheless, my experience has been that this is probably the most de-emphasized method of all the primary factoring methods (common factor, difference of squares, trinomial type, sum/difference of cubes). At least that was the case in the several dozen college algebra and precalculus classes I taught from the mid 1980s to the mid 2000s. I taught it, but it wasn't something that would prevent a student from getting an $A^-$ or an $A$ on a factoring and basic equation solving test if they pretty much knew all the other types reasonably well. $\endgroup$
    – Dave L Renfro
    Nov 5, 2020 at 21:55
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    $\begingroup$ In my comment I had to be a bit concise. There is reinforcement of certain skills and concepts, in which I would include the ability to go from recognizing that $x$ is a common factor of $3x^2 - x$ to being able to recognize that $x-2$ is a common factor of $3(x-2)^2 - (x-2)$ (a fairly large leap for most students), as well as additional practice in routine algebraic manipulation and number sense (which aren't routine until practiced one way or another until they become routine), and actually needing to use such factorizations for something else. $\endgroup$
    – Dave L Renfro
    Nov 6, 2020 at 0:23
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    $\begingroup$ @MikePierce The reason the pattern-matching works is factoring by grouping. I think I would argue that the thing you are doing is factoring by grouping, you just aren't telling your students that. This question reminds me of "why would I teach completing the square when I have the quadratic formula?" (Which is also a good question). $\endgroup$
    – Chris Cunningham
    Nov 6, 2020 at 1:53

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That is a great question that I struggled with the one year I taught Algebra 1. I came to the same conclusion that factoring by grouping problems are inauthentic because they only work when the expression is built to work.

The story that got me through it is that you can use factoring by grouping to factor a quadratic expression if you manage to come up with a clever decomposition of the linear term that makes it work. (One of my main points of the unit is that factoring polynomials is the inverse of multiplying polynomials, so we want to find a strategy to undo the "box method" of multiplying binomials and collecting like terms.)

So what is the clever decomposition of, say, $x^2+7x+12$? Here (and since I was working with non-accelerated ninth graders I just told them that this was the trick rather than formally justifying it), the trick is to think of two numbers that add to 7 and multiply to 12. These numbers are obviously 3 and 4, so we can decompose the expression into $x^2+3x+4x+12$ and use the factoring by grouping strategy to come up with a factorization of $(x+3)(x+4)$. So, at least for me, factoring by grouping is a "lemma" strategy that lets me solve the far more authentic problem of factoring quadratic expressions.

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This is a thought provoking question. Learning to think algebraically is a gradual process. It takes practice to see that $$ax+ay=a(x+y)$$ is the same as $$(x+1)x+(x+1)y=(x+1)(x+y).$$

Eventually, some students will have enough algebraic fluency to work with something like $f(x+h)$ where $f(x)=x^2-x.$ Such a successful student will have developed fluency parsing algebraic expressions where the "atoms" are not just single-character variables such as $a, x, y$, but potentially more involved expressions such as $x+1$ or $x+h$.

Factoring by grouping is not a means to an end. If our sole purpose was to have students factor expressions, we could just teach students to use computer algebra system software, and we could ignore the goal of developing algebraic fluency.

An analogy might be a child learning to read. The beginning reader can parse and write simple sentences such as "The dog runs fast." Some years later, the student will be like George Bernhard Shaw, writing complex sentences such as "A life spent making mistakes is not only more honorable, but more useful than a life spent doing nothing.”

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While it's pretty clear from your question what method you're talking about, the title isn't so clear. When I hear factoring by grouping, I think of something like $3x^2+2x+9x+6$, and not $3x^2+11x+6$. When the student has to first come up with a way to split the middle term I've heard it called, not surprisingly, factoring by "splitting the middle term" and also "the AC method".

As for the AC method, I agree with Dave L Renfro's comment that one reason for teaching certain things is to reinforce basic skills. A little practice with some mental arithmetic is never a bad thing, although I guess they'd get that with pattern matching too. AC also includes what I would call factoring by grouping in the final steps, so that gets practiced as well. I've personally taught the method before because I had the time and freedom to do so, and because I think it's a cool method. However, if I had to choose between AC and, say, completing the square, I think I'd teach completing the square every time. AC is nice but omitting it is no big deal.

For what I would call factoring by grouping, as in my first example $3x^2+2x+9x+6$, I would want all students to be able to factor this by inspection. There's at least one major result down the road that it's very useful for: proving the product rule by writing $$f(x+h)g(x+h)-f(x)g(x)=f(x+h)g(x+h)-f(x+h)g(x)+f(x+h)g(x)-f(x)g(x)$$ which is itself a great example of proof by "hey this term might be helpful to have so I'll just jam it in there and make up for it elsewhere to keep the equality". (Another great example is completing the square, which doesn't have as much of a rabbit out of a hat feel as the product rule.)

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  • $\begingroup$ Actually, I wasn't thinking specifically of the AC-method (although it makes use of factoring by grouping), but rather something like $x^3 + 2x^2 - 3x - 6 = x^2(x+2) - 3(x+2),$ where one then factors out $(x+2).$ In applications, one-variable examples like this show up rarely and to me the relatively little attention paid to it seemed almost entirely for the purpose of providing a different kind of algebraic practice. Indeed, the later-studied (at least used to be; was getting phased out in many U.S. classes even in the 1990s) rational root theorem more than takes care of later applications. $\endgroup$
    – Dave L Renfro
    Nov 7, 2020 at 4:50
  • $\begingroup$ Factoring by grouping becomes more applicable in multivariable factoring, and despite the fact that this method will virtually never work on a randomly chosen multivariable polynomial, it is a useful tool for those polynomials that do factor in nice ways. But probably not that important unless someone is wanting to prepare for math contests, perform well on certain tests (comes up occasionally even on this test), or planning to teach high school or early college math (where knowing everything like this is important, even if not taught). $\endgroup$
    – Dave L Renfro
    Nov 7, 2020 at 4:59
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    $\begingroup$ Another one I use all the time: $\frac{2x+4}{x+7} = \frac{2x+14-10}{x+7} = \frac{2(x+7)-10}{x+7} = 2-\frac{10}{x+7}$ $\endgroup$
    – Steven Gubkin
    Nov 7, 2020 at 12:52
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    $\begingroup$ @Steven Gubkin: I've seen people who like this method, which isn't "guess-and-check" (what some think when first seeing it) but rather can be done in a systematic way, but for some reason I find it simpler to do the long division (one simple "division step" needed, after which you're at the "remainder stage"). For those who don't know why one would do this . . . can now immediately integrate, can now recognize the graph as an appropriate vertical-shift/horizontal-shift/reflection/dilation of the hyperbola $y = \frac{1}{x},$ can now easily obtain power series using geometric series ideas, etc. $\endgroup$
    – Dave L Renfro
    Nov 7, 2020 at 16:13

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