Polynomial Division

Sunday 19 June 2005 at 2:50 pm | In Articles | 7 Comments

A long time ago UK students used to learn how to do polynomial long division before they were 16. Nowadays, they see little of it until they reach A level, and the new national syllabus expects them to learn the remainder theorem for AS (16+). They don’t actually need to do long division as examples are usually simple enough to allow one to guess factors; for example x^3-7x+6=(x-1)(x^2+ax-6) and a has to be found. Nevertheless, it is useful to be able to do long division and, given the time constraints, I usually use Synthetic Division.

\LaTeX allows you not only to show the steps of either method, it will also do the mathematics for you, thanks to the polynom package. The code \polylongdiv{x^3-7x+6}{x-1} produces


and \polyhornerscheme[x=1]{x^3-7x+6} gives


There’s lots more possibilities such as


Brilliant! You can watch an online demo of the polynom package doing division step-by-step in a number of different ways here. Click to move the demo on, press Esc to end it.

Please note: \polyhornerscheme is not available in versions of polynom before version 0.16, so if you wish to use \polyhornerscheme do make sure you get the latest version, perhaps from here


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  1. When people find out that I’m studying maths they always go, “No way! Can you do long division, I’ve never understood that,” and when I reply “Hey, I can long division with x’s and y’s,” they are well impressed!

    God I’m soooo cool.

    I can’t remember where I learnt it, perhaps on the further maths A-level?

    Comment by Craig — Sunday 19 June 2005 6:54 pm #

  2. So few students are studying further maths these days that the government is funding a project that will allow schools and colleges to get together to teach students where there aren’t enough to run classes in a particular school or college.

    It’s a sad indictment of the state of affairs of mathematics these days.

    Comment by Steve — Sunday 19 June 2005 7:33 pm #

  3. That’s really bad, I can’t imagine I’d ever had gone on to study maths at degree level if doing Further Maths hadn’t piqued my interest. Moreover, the first year of your degree is made a whole lot easier if you’ve done some complex numbers etc. before…

    Comment by Craig — Monday 20 June 2005 8:10 pm #

  4. Hi, I am pretty new at latex, so maybe my question is stupid….sorry if it is the case. I am trying to use \polyhornerscheme[x=1]{x^3-7x+6} but I keep getting !undefined control sequence. Do you know what could be wrong and what I need to do to use this command? thanks in advance for your help, Aude

    Comment by aude — Wednesday 24 August 2005 1:35 pm #

  5. You usually get that error if you have mistyped a LaTeX command. Try reducing the LaTeX code in the document until you find the source of the error.

    Does this minimal example work?


    Comment by Steve — Wednesday 24 August 2005 2:29 pm #

  6. Aude
    The problem could be that you have an older version of polynom which doesn’t support \polyhornerscheme. Try changing to the latest version here

    Comment by Steve — Sunday 9 October 2005 11:45 am #

  7. What about if you want to display polynomial long division in modular.., for example, I am trying to long divide a polynomial in mod7

    Comment by John P — Friday 23 April 2010 1:14 am #

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