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. […] 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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