Jul 21

Polynomial Division – Long, Synthetic or Boxes?

If you read my last post, then you know that the blog name here came out of a student comment made right after I finished a lesson that I was particularly excited about.  This is it!  While it might not seem like much here – trust me, I was excited.  I had that “ah-ha” moment that I hope to give my kids.  Love it.

My Algebra 2 kids had been working on factoring and finding roots of higher-order polynomials.  We had done the typical “what do you notice?” with the graphs of the polynomials, talked about types of roots and how we might write the factors and find the roots – both real and complex.  We were coming to the end of the unit and most kids understood the why and how of the process but some were still having trouble with the division.  We had practiced both long and synthetic division.  I had lead them through discovering that they could use synthetic division given any root – even irrational and complex (yes – that’s true – never say never to a kid, they’ll prove you wrong!  Math teachers will too – see Never tell me it can’t be done.)  They were still struggling – so I was trying to think of a new method.  Then the epithany….

We use boxes to multiply – why not divide?  So, I searched online to see if I could find it.  No luck (I stink at searching – found these later – examples and mathrecreation)  So, I tried a few.  Guess what?  It worked… and seemed pretty easy.  Here are some examples.


Download (PDF, 124KB)



What I really liked about this method was that the kids had to connect the multiplying and dividing.  They had to think through it.  It doesn’t feel as algorithmic as regular long division.  Thoughts?

Question – could we apply this in some way to completing the square?  The other Algebra 2 topic that makes me crazy!


What I love about twitter – sent a tweet out about this post and in no time at all had an answer to my question!  Bob Lochel (@bobloch) read my post and responded in his blog, mathcoachblog.  And, of course, I had a “well duh!” moment.  Please drop over there and read his blog!  It makes so much sense and is a much more visual representation than the typical algebraic process seems to be.  Thanks Bob!


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  1. Kathryn

    Just found your blog through Twitter! Welcome to the math blogging world.

    Thanks for sharing this method. It’s new to me and I think my students might like it, however they have done well with the other methods in the past as well. I’m not sure about applying this to completing the square. We did a lot of work with Algebra tiles on that prior to introducing the algorithm and this seems to be a natural bridge between the two. I don’t know how successful kids would be if the numbers weren’t “pretty” for completing the square this way.

    1. lbburke

      Thanks so much for your comment Kathryn! I’m new to blogging and am really doing it for myself but its nice to think that others might want to read it too.
      My kids really liked this method but each ended up picking their favorite – which is what I like. As a student, I always wanted to have some choice.
      I, too, use Algebra tiles to help introduce completing the square. I also just use some pattern work to help students see the connection between the vertex form and the standard form. They are pretty good at seeing the pattern and then developing the algorithm from there. Again, just always like looking for a new way to get that “ah-ha” moment from a kid.
      Thanks again!

      PS- Read your blog all the time too (don’t look too closely at my grammar though!)

  2. J.P. Brichta

    Nice one! We have been doing polynomial division like this for years. This graphical representation of the distributive law builds directly from our use of manipulatives (algebra tiles) in Grade 9, factoring polynomials in Grade 10 and 11 (using a chart), and finally division of polynomials in Grade 12. In fact, I understand our elementary panel teaches multiplication of two multi-digit numbers in a similar way.

    1. lbburke

      Thanks! I thought I was being so original at first (part of the excitement) but I did find other examples online. Still didn’t take away from my ‘ah-ha’ moment though. Kids loved it.

  1. Completing the Literal Square | mathcoachblog

    […] interesting post comparing polynomial division methods on the More Than a Geek blog reminded me of my own experience teaching completing the sqaure. […]

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