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Why do people think long division of polynomials is complicated ?

I heard this expressed recently and it seems like an odd sentiment. For me, synthetic division is complicated and totally adhoc because it looks nothing like the division of integers I remember from my childhood. In contrast, long division of polynomials is just the same process I know and love from the arithmetic of my youth. To me, long division of polynomials is just a natural extension of long division of numbers.

I generally refuse to cover synthetic division because it's not significantly faster and it fails to cover division by nonlinear polynomials. Why teach something which is weird and not general. Just to make that particular calculation faster ?

In addition, long division of polynomials requires multiplication of polynomials in the calculation. This is a skill which needs reinforcement. In contrast, synthetic division is a one-way street of shifting numbers around without any particular attachment to the larger discussion of polynomial algebra.

What am I missing ?

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    $\begingroup$ For what it's worth, I feel the same way about synthetic division. Despite using it a lot in high school precalculus math (1973-74) and teaching it many times since then, if I haven't done it for 2 or 3 years, I have to review how it goes (or do a couple of trial runs for polynomials that I know how to factor, such as $x^3 - 1$ and $x^3 - 3x^2 + 3x - 1).$ The only thing I can think of where it is reasonably helpful is when you're using the rational root test (many times), which is something probably not much taught anymore, at least not as thoroughly as it used to be. $\endgroup$
    – Dave L Renfro
    Commented Oct 27, 2021 at 15:05
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    $\begingroup$ Long division is complicated, polynomial or not. $\endgroup$
    – candied_orange
    Commented Oct 27, 2021 at 17:52
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    $\begingroup$ @SueVanHattum mathworld.wolfram.com/SyntheticDivision.html gives an example of dividing $4x^5+x^3-3x^2+2x-7$ by $2x^3-x+1$. It's ... a journey. MathWorld makes the claim that what we're calling synthetic division is the most elementary subcase and we should all be calling it Ruffini's rule instead. $\endgroup$
    – Matthew Daly
    Commented Oct 27, 2021 at 20:09
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    $\begingroup$ @JamesS.Cook: I suspect that it's taught at some point, but not really verified that students digested it. I see in the Common Core standards (FWIW) at Grade 6 they should "Fluently divide multi-digit numbers using the standard algorithm", then at Grade 7 they should, "Convert a rational number to a decimal using long division". Totally agree with your other comments (e.g., Grade 6 is in the danger zone with non-specialist teachers not knowing what they're talking about, or why it's important later, e.g., for polynomial division). $\endgroup$
    – Daniel R. Collins
    Commented Oct 28, 2021 at 14:57
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    $\begingroup$ AFAICT (from personal memory and this Google Scholar search), in England and Wales, long division of numbers has gone through two cycles of being removed from the school curriculum (the first time c. 1975, the second c. 2000) then reintroduced (the first time c. 1988, the second c. 2014); whereas long division of polynomials has always been on the curriculum (at A-level). $\endgroup$
    – user14522
    Commented Oct 28, 2021 at 16:20

5 Answers 5

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I like to include synthetic division as a topic in a college algebra or precalculus course. It is an opportunity to take a 20 minute digression to talk about Horner's method, which is used in several computer science algorithms.

It is a clever way to evaluate a polynomial that improves the naive approach of evaluating all the terms which might be large numbers (appearances of $c_k 10^k$), and then adding all these evaluated terms, which might include subtractions of large integers. This sort of perspective can begin opening the eyes of some STEM students to issues related to numerical algorithms. It might put some to sleep. But it excites me, so they wake up! It is the unique day of the semester when I demonstrate using Matlab.

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    $\begingroup$ I was just gearing up to talk about Horner's algorithm. If nothing else, it reduces the number of multiplications required to evaluate a polynomial from $n^2$ish to $n$ish. $\endgroup$
    – Xander Henderson ♦
    Commented Oct 27, 2021 at 14:40
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    $\begingroup$ This appears to have nothing to do with the question that was actually asked here. $\endgroup$
    – Eric Wofsey
    Commented Oct 27, 2021 at 20:45
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    $\begingroup$ Anecdote: I actually lost a job once due to knowing Horner's method and implementing it in a coding test -- because the interviewer couldn't believe it would work (even after I explained & hand-traced it for them, etc.). That time I didn't get the job. $\endgroup$
    – Daniel R. Collins
    Commented Oct 28, 2021 at 1:36
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    $\begingroup$ @DanielR.Collins, what, I thought that was the common-sense implementation. I'm pretty sure I've seen it done by others too, and never even thought it had a fancy name :D $\endgroup$
    – ilkkachu
    Commented Oct 28, 2021 at 13:26
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    $\begingroup$ @DanielR.Collins: If a programmer is unable to visualize what sum * 10 + digit is doing (appending digit to the base-10 representation of sum), then they have no business implementing numerical algorithms. $\endgroup$
    – Kevin
    Commented Oct 28, 2021 at 22:39
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(This answer is me speaking as a student. I taught both polynomial long division and synthetic to my first calculus class, but realized by the end of the unit that it was really unneeded for AP questions and didn't teach it again.)

So as a person who factors polynomials every once in a while, I don't like the large number of sign flips polynomial division requires for the subtraction phase and the fact that you only need to misread one to get the wrong answer. I also prefer the reduced writing space that synthetic uses. (When I learned both techniques back in the day, we were asked to factor cubics and quartics with the rational root test, so there was plenty of scratch paper used up for trial and error.)

I get that synthetic looks artificial in the context of numeric long division. But, TBH, that's a pretty artificial manipulation of figures itself. I wonder if synthetic would look more attractive to you if your teachers had switched you to short division once you were ready for it....

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    $\begingroup$ Maybe, but they didn't, so I prefer the long division. Also, as I said, long division reinforces the process of polynomial multiplication. And, being careful with signs is a worthwhile endeavor. In any event, thanks for adding the comments about the more general aspects of synthetic division. I had tried to find that for my post, but there is just an unending sea of algorithmic tutorials I could not get past. $\endgroup$
    – James S. Cook
    Commented Oct 28, 2021 at 7:08
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I think the fear of poly long division is based on the amount of symbol manipulation and writing. So, I get how the students feel. But completely agree with you that this needs reinforcement.

For synthetic division, I don't remember ever using synthetic division in any later math or science classes. (To the extent, I don't recall learning it. Probably did, went to very good school...but maybe even didn't.) Maybe synthetic division is a little like the tic-tac-toe (tabular integration) shown in Stand and Deliver. Kind of a cool flourish and worthwhile if you do a gazillion problems. And does take some of the pain of writing and symbol manipulation away. But still more important to learn/understand the long way as a base.

In a time constrained environment and with weaker kids, I would pitch the synthetic division. It's pretty optional. Like I said, don't recall using it later in any other course. I would keep the poly long division since it's kind of analogous to manipulation done later in partial fractions and that gets used in lots of other courses.

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    $\begingroup$ I have no idea why this is downvoted, the amount of writing was the same reason we didn't like integer long division. I actually remember this part of gradeschool because my teacher showed us a shorthand for doing the same math but with a lot less writing, and it was a lot less painful to go through the worksheets for all of us. $\endgroup$
    – Izkata
    Commented Oct 27, 2021 at 18:34
  • $\begingroup$ Great googly moogly, I think I agree most with guest at the moment. $\endgroup$
    – James S. Cook
    Commented Oct 28, 2021 at 7:14
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How did you learn to multiply 2 or more digit numbers?

If you learned "long multiplication", then so be it.

But if you learned the grid method, then synthetic division links right back to that. In synthetic division, you write out the grid and fill in the gaps to find "what do you need to multiply by the divisor to get the polynomial in the dividend."

This means if you understand why grid multiplication works (which is easier to explain than long multiplication), you can understand why "synthetic division" works.

You can teach this division method as just shifting numbers around. But the strength in the method is not in how quickly students can find an answer, but in the connection that they can make to grid multiplication and the notion of division as the inverse of multiplication.

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    $\begingroup$ The grid what now? $\endgroup$
    – Daniel R. Collins
    Commented Oct 28, 2021 at 1:43
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    $\begingroup$ So it is connected to a multiplication method which is not widely known ? So, that wouldn't help me teach it, I'd still have to teach the class this novel method of multiplication. $\endgroup$
    – James S. Cook
    Commented Oct 28, 2021 at 7:12
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    $\begingroup$ @JamesS.Cook: As an American, I understood this method within about five minutes of opening its Wikipedia article. It does not strike me as a particularly complicated or weird thing to teach. $\endgroup$
    – Kevin
    Commented Oct 28, 2021 at 18:56
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    $\begingroup$ How exactly does grid multiplication and synthetic division go together ? I would like to see some details on this claim. $\endgroup$
    – James S. Cook
    Commented Oct 28, 2021 at 22:23
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    $\begingroup$ @JamesS.Cook: If the problem is that students have no number sense, isn't that the problem? It has nothing to do with long division. If students do not grasp why $a/b = c/d$ implies $a·d = c·b$, then the bad terminology "cross-multiply" only makes things worse. Same for "long division" and "synthetic method" and whatever other method people might think up. $\endgroup$
    – user21820
    Commented Oct 29, 2021 at 13:43
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This is a perspective from an adult who self-studies math. I felt this way about synthetic division up until recently, when I sat down to actually learn synthetic division. But then, once I got it, it made perfect sense why people like it. It's less mentally taxing once you get a handle on it.

If you say "why should they care about that?", then I direct you to the quadratic formula. Why do you teach it? You can always complete the square. But we don't. Why teach the power rule in calculus? You can always derive it just as fast from the definition of a derivative rather quickly. But we teach it.

Students generally feel like, once they get beyond a certain part of math, they just want to be able to get it done quickly without a lot of mental effort or space. The division isn't the main point of what they're doing, so they want to get it over with and back to the fun part of math.

My grandpa was a math teacher, and he would teach me a lot of shortcuts, including this weird way of multiplying by multiple of 5. I know how to do square roots longhand, even though Newton's method is right there. Mathematicians love their shortcuts, too.

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