Synthetic Division: to teach or not to teach?

I really can’t think of a good reason why synthetic division is still in the Algebra 2/PreCalculus curriculum.  When I was in high school (at my HS, I was in the last class not to use graphing calculators), synthetic division was a handy shortcut because we had to do tons (often enough for a whole page for one problem) of divisions to test potential zeros for higher-order polynomials.  However, with technology as it is, I can think of better ways to spend the 2-3 days it would take to cover synthetic division and finding rational zeros by hand.

I do still teach polynomial division, because it is useful for finding polynomial asymptotes of “improper” rational functions.  But for the synthetic division shortcut, unless I’m doing tons and tons of polynomial division problems (and only the case where we’re dividing by linear factors at that), I really can’t think of a good reason to take the time to do it.  Plus, I have a really bad memory and always have trouble remembering the proper sign for the number on the outside of the synthetic division box (r when the divisor is x-r…).

At my school, synthetic division is taught in Algebra 2, so I’ve been able to use that reason to skip it in PreCalculus.  However, our local community college lists it as a topic that is taught and covered on the final in PreCalculus.  So, if I have students who opt to do the college in the high school program, I do need to cover it in PreCalculus…

Please let me know if there’s something I’m missing, or why you do or don’t do synthetic division.


About Clint

Looking for answers and enjoying the journey.
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11 Responses to Synthetic Division: to teach or not to teach?

  1. samjshah says:

    Personally, I don’t really care much for it [synthetic]. Polynomial division doesn’t take a long time, and shows exactly what’s going on. Synthetic division, I find, obscures things. The one good question that synthetic division raises for me is: explain why this is equivalent to polynomial division.


  2. mythagon says:

    I’m not a huge synthetic division fan myself. I learned it the first time a few days before teaching it my first year and I feel like I have to relearn it every year whereas polynomial division is very straight forward for me.

    The only part I do like about it is how it brings back the idea of place-value with respect to x’s now instead of # of 10s. And I like Sam’s question above in regards to the place-value mindset. I also enjoy showing students how familiar things can be abstracted and broken into the simplest bits before solving.

  3. Clint says:

    Thanks for the feedback – those are some good questions and connections to bring in if I’m forced to cover it this year.

    I do like polynomial division, but my first year I made the mistake of assuming that students knew long division. After that, when I’ve asked, I found that I had pockets of students that had never seen long division before. So I’ve used it as an opportunity to talk about how the long division algorithm works – it’s one of those topics where I use the “imagine we need to help our younger sibling/cousin/friend with their homework” approach, then move into doing polynomial division.

    • Jeremy Touzel Hansuvadha says:

      Funny, I always get confused with the sign, too.

      I agree with Sam–is synthetic division that superior to polynomial division? It is faster, yes, but is that savings in time worth the trade off? Heck, it was actually probably slower for me some of the time, when factoring in the time it took to do the scratch work necessary to determine how the signs went.

  4. Given that I probably don’t remember how to do it, and can’t imagine a situation where I’d need to do it while denied access to technology or other methods, I’d have to say my opinion of synthetic division is pretty low. My #1 non-tech method is definitely using a rectangular area model (Ever seen it? It’s pretty cool.) or I could give regular polynomial division a shot.

  5. AA says:

    Synthetic division is an automated process that can be used in a computer programme. This is a big advantage. One cannot perform polynomial division in programming, which is a manual process.

  6. cc says:

    I agree with the last comment. The only “theoretical” advantage of synthetic division over polynomial long division is that if you choose to emphasize algorithms in your class, one can more easily imagine writing a computer program to perform synthetic division than for performing polynomial long division.

  7. Victor says:

    Ummm….computers can also be programmed to do polynomial arithmetic. It’s nit that big a deal. So I don’t think this last point is really valid. I just had to remind myself about how this worked to help my daughter with her homework. I think it makes much more sense to just stick with long division.

  8. Reblogged this on Shade Tree Math Teacher and commented:
    One person asks, Why teach synthetic division? and I’ll see that question and raise you a “Why teach polynomial division?”

  9. I THINK its better the synthetic division for solving division. I rather use it than long division

  10. Bill says:

    another question. Why test on it? I hope it is not included in standardized testing.

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