26 May 2015

Algebraic Division

During public exam season I'm afraid I can't resist looking at The Student Room - not to read all the infuriating student posts where they obsess about lost marks and grade boundaries, but simply to see each exam paper as soon as possible. I have no patience. Arsey, presumably an anonymous teacher, posts model answers very soon after the exams have taken place (in previous years not until the next day, but now there are different papers for different time zones the model answers are posted straight after the UK sitting). When I was reading Arsey's recent C2 solutions, I noticed his method for polynomial division:
I've seen him do this before. It always occurs to me that this is far more straightforward than the long division method I teach my students. Why do I teach long division?  I suppose it's because it features in the C2 textbook so I've always assumed it's the 'best' method. Perhaps next year I'll try an alternative. In this post I look at four methods for polynomial division. 

1. Long Division
This method often features in A level textbooks. It just involves following a series of steps (divide, multiply, subtract, bring down, repeat) - an algorithm learnt by drill rather than through understanding. The steps are familiar to those who learnt long division at primary school. Those who were taught alternative division methods (eg chunking) are at a slight disadvantage but do catch up quickly. Practice makes perfect. This isn't an elegant method - it's totally procedural and isn't particularly nice to teach, and students have to know special rules for special cases (eg including a 0x term) - but it does the job just fine. It works well when there's a remainder.
2. Grid/Box Method
I've just tried this method for the first time and I can't believe how easy it is - and so much quicker than long division! All you have to do is set up a multiplication grid - start by filling in the bits you know and then the rest follows by logic. This video from Bon Crowder explains the method very clearly. James Tanton calls this the Galley Method - his Curriculum Essay about how it works includes exercises and interesting questions. The picture below from Esther (@MrsMathematica) shows a sensible layout which makes dealing with remainders very easy.
3. Inspection
This is like the grid method but set out differently. All you have to do is write your polynomial as the product of a linear function and an unknown quadratic (or cubic, quartic etc, depending on the question) then use logic and algebra to work out the numbers by equating the coefficients. It's quick and fairly straightforward. It's also easier to follow what's going on than in the confusing algorithm of long division.
In a 1940s textbook I spotted an alternative layout for inspection that I love. In the example below we want to divide by x - 1 so we write (x - 1) three times and then just fill in the rest. Again, it's quick and logical, and easy to keep track of each term.
4. Synthetic Division
I don't like this method so I don't really want to mention it here, but for completeness I suppose I should. I've found lots of (often negative) reference to it on American websites but I've never seen it used in England. I'm told that it's commonly used in Scotland (thanks to @mrallanmaths, @kenniejp23 and Paul Smith for commenting). The reason I dislike it is it appears to be one of those 'remember the steps but have no clue what's actually happening' methods.
Source: acedemic.utep.edu
I followed the rules above and it did work, plus it was pretty quick, but it was a bit like doing a magic trick.

The animation below shows the equivalence of long division and synthetic division. It looks to me that synthetic division is just a confusing method made even more abstract. There are many defenders of synthetic division though - they say that it's an acceptable method providing students are taught the underlying concept before they start applying the super-efficient algorithm.

Source: purplemath.com

So that's it - four methods for dividing polynomials. This PowerPoint from the Further Mathematics Support Programme summarises the first three methods.

Which do you prefer?


  1. You are right about it being Suduko like- I don't really know much about polynomial a at all but now I can enjoy dividing them!

  2. Synthetic division is used in the UK - in the Scottish Higher papers and textbooks. I, too prefer the box method (which is just formalised inspection), but synthetic division, like long division, can also deal with remainders easily.

    1. Thanks Paul. I had no idea it's commonly used in Scotland. I've updated the post to reflect that. It's funny that there are geographical differences in mathematical methods.

    2. In Argentina we use the synthetic method, it's called "Ruffini's rule" and it appears in textbooks as well. Commonly used in Secondary and University.

    3. Thank you! Very interesting. It's used more widely than I realised.

    4. It's used on the continent too. I was taught it by a Belgian student, called the Horner method, I believe. And I like it and also now teach it to students. Also good for remainders.

  3. Great post, Jo. I learned the long division way at school but a few years back came across the box method which I much prefer (as do my students). I'd never heard of synthetic division until I came to Canada. Most kids don't like it. I think it is shown because it is 'quick' (whatever that means) but I have managed to avoid it.
    btw, don't you just love James Tanton's stuff?

    1. Yes, I love James Tanton's stuff, it always gets me thinking. His work on mathematical methods is unique and much needed by maths teachers.

  4. Synthetic division is typically taught in the second course on algebra (typically called Algebra 2) in US high schools. I usually teach honors, so I ensure my students understand polynomial long division before I show them synthetic division. However, I know there are other teachers that teach synthetic division to those students who simply don't get long division.

    1. That's interesting, thank you. Unfortunately there are many topics in which students who don't get it are taught a shortcut instead. I'm guilty of it myself sometimes. It achieves one aim (higher test scores) but there's no underlying understanding of mathematics. Here I think teachers should perhaps try the grid method instead of synthetic division.

  5. The box method is great but only works for the factor theorem. The advantage of synthetic division (and long division) is that it works for the remainder theorem as well.

    1. The box method can find remainders as well. If it doesn't divide perfectly, you can find the remainder.

      I love teaching the box method, kids understand it so much easier than long division.

  6. I love 'by inspection'. It feels logical and no-fuss (less writing!), and it still works when there's a remainder.

  7. Enjoyed this post! It's interesting to see the grid/box method, but I don't know if I'd actually use it with students - nearly all that I've taught are perfectly happy with the long division algorithm, and then those that can do "by inspection" just don't bother with the algorithm.

    The first time I learned long division was dividing polynomials at A Level - my teachers at KS3/4 had completely neglected to teach it!

  8. Hi,
    I've always taught the long division method because it relate directly to....long division. Trouble is despite teaching this to A level students, they can't actually do Long Division!! However, last year I started teaching the inspection method, with success but one student found the Synthetic method on the Khan Academy web site and students seem to love it. I hate it! I also found students in Zambia keen on this method as it is taught there. This year I although I teach the Long Division (old dogs/new tricks) I show the box method and the simplicity of it and students prefer it.

  9. I found another method! Grouping! http://www.wikihow.com/Factor-a-Cubic-Polynomial

  10. I feel like when I say I'm doing this 'by inspection', I don't mean the same as you do here. I think I mean 'long division without the bus-stop'.

    Say I was dividing x^3 + 6x^2 + 8x + 3 by x+1.
    First I write (x+1)(
    Then I decide what I need to make x^3: (x+1)(x^2
    And I write down nearby everything I've made so far: x^3 + x^2.
    I've not got enough x^2 yet, so I'd better put another 5x in the second bracket: (x+1)(x^2 + 5x
    Getting close, I've now made x^3 + x^2 + 5x^2 + 5x. So I still need another 3x etc.

    If you took a photo of my work when I'm done it would just look like:
    x^3 + 6x^2 + 8x + 3 = (x+1)(x^2 + 5x + 3) with x^3 + x^2 + 5x + 3x + 3 written somewhere nearby!