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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 ?

James S. Cook
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  • synthetic is also possible for higher degree polynomials, but I don't think that students fear long division – YOu will not know Oct 27 '21 at 05:15
  • @AdityaDwivedi, can you link a demonstration? My understanding it that the divisor must be linear. – Sue VanHattum Oct 27 '21 at 14:44
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    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. – Dave L Renfro Oct 27 '21 at 15:05
  • One thing I noticed, when I was learning long division: for most of the process, I am doing multi-step algorithms while there is an incomplete number on top of the bar. There's something odd about seeing a "5" above the bar, and realizing it isn't a number, but a shorthand for "fifty-something, and we don't know the 'something' yet". It gives me a feeling of a lack of closure. The only way to really rationalize it, that I know of, involves treating it as a string of digits. I have found historically this gives many people great grief. Just try to teach them another base, like binary. – Cort Ammon Oct 27 '21 at 17:44
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    Long division is complicated, polynomial or not. – candied_orange Oct 27 '21 at 17:52
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    @SueVanHattum https://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. – Matthew Daly Oct 27 '21 at 20:09
  • I saw this in the Hot feed and could not help wondering what is the proper ___phobia name for fear of polynomial long division. I wonder if it can be comorbid with Triskaidekaphobia. – JDługosz Oct 27 '21 at 22:54
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    "the same process I know and love from the arithmetic of my youth" -- most people now don't know how to long-divide integers. A process dependent on that is a non-starter. – Daniel R. Collins Oct 28 '21 at 01:34
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    @DanielR.Collins that is just sad. We don't teach long division of numbers now ? No wonder students can't add fractions or do much any arithmetic unless I wait a silly amount of time. Whatever is going on in the elementary schools, it's a disaster. – James S. Cook Oct 28 '21 at 07:17
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    Funnily enough I was NEVER taught long division at school - we went straight to short division. So when long division of polynomials came up and they told me it was just like long division I learnt at school I was a bit lost. – Muzer Oct 28 '21 at 13:31
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    @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). – Daniel R. Collins Oct 28 '21 at 14:57
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    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). – Daniel Hatton Oct 28 '21 at 16:20
  • @SueVanHattum - I found this https://www.purplemath.com/modules/synthetic-division-sgnd.pdf and need to look at it more, but it got my attention. – JTP - Apologise to Monica Oct 29 '21 at 09:55
  • FWIW, I have the sentiment that polynomial division as an algorithm is outdated... – Lenny Mar 12 '24 at 23:03

5 Answers5

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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.

user52817
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    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. – Xander Henderson Oct 27 '21 at 14:40
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    This appears to have nothing to do with the question that was actually asked here. – Eric Wofsey Oct 27 '21 at 20:45
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    @Eric Wofsey while the question started off asking about fear of polynomial long division, it developed to a question about why synthetic division should be taught, presumably in an algebra/precalculus course. – user52817 Oct 27 '21 at 21:14
  • @user52817: Synthetic division appears in the context of why it should be taught as opposed to polynomial long division. Nothing in your answer seems particularly relevant to that, since polynomial long division can be used as an evaluation method in exactly the same way. – Eric Wofsey Oct 27 '21 at 21:32
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    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. – Daniel R. Collins Oct 28 '21 at 01:36
  • Given my interest in computer science... this would be a bug not a feature of synthetic division. More power to you, just not my thing. – James S. Cook Oct 28 '21 at 07:05
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    @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 – ilkkachu Oct 28 '21 at 13:26
  • @ilkkachu: Right? Exact same for me at the time! But multiple times when I've shared this anecdote I've gotten a response of, "I wouldn't allow that in our codebase, it's too complicated for normal programmers to understand". – Daniel R. Collins Oct 28 '21 at 17:15
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    @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. – Kevin Oct 28 '21 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....

Matthew Daly
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    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. – James S. Cook Oct 28 '21 at 07:08
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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.

guest
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    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. – Izkata Oct 27 '21 at 18:34
  • Great googly moogly, I think I agree most with guest at the moment. – James S. Cook Oct 28 '21 at 07:14
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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.

Glorfindel
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James K
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    The grid what now? – Daniel R. Collins Oct 28 '21 at 01:43
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    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. – James S. Cook Oct 28 '21 at 07:12
  • It is a standard method of multiplication in the UK – James K Oct 28 '21 at 08:30
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    @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. – Kevin Oct 28 '21 at 18:56
  • @Kevin what makes it weird for most simple minded folks is the striking non-uniqueness of the method. I understand it, in fact that's more or less how I multiply in my head in lecture (but I just use a row typically). In contrast, the standard algorithm is an algorithm. There is no adhoc splitting of factors at the outset of the calculation. In my view, that step alone makes the grid method far more complicated for students who have no number sense. – James S. Cook Oct 28 '21 at 22:20
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    How exactly does grid multiplication and synthetic division go together ? I would like to see some details on this claim. – James S. Cook Oct 28 '21 at 22:23
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    @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. – user21820 Oct 29 '21 at 13:43
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    @user21820 my point is that the regular algorithm for multiplication does not require number sense. Good or bad, that means students who have no sense on how to split the number for the grid method may still be completely competent on the regular algorithm. No creativity or cleverness is needed, just the multiplication tables. So, in that sense the regular algorithm is far better than the grid method. It does not require students to start with an adhoc choice on how to split the number. Should students learn how to make adhoc choices ? Sure. Should we hang our hat on such an adhoc method..no. – James S. Cook Oct 29 '21 at 19:53
  • This is drifting very far from the topic... but the whole point of mathematics education is to understand. Learning without understanding is not just bad teaching, it is bad learning. – James K Oct 29 '21 at 20:52
  • @JamesS.Cook: If the regular algorithm for multiplication is not taught with clear explanation so that students really understand how it works, then it should not be taught at all. I agree with James K; lack in understanding, no matter for which algorithm, is just a recipe for bad teaching and bad learning. If you don't believe me, watch any student who lacks basic mathematical understanding compute 7999×6998. Regular algorithm, yea? Lol. – user21820 Oct 30 '21 at 03:03
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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.

trlkly
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