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Today, mid-polynomial unit, I taught a class polynomial long division and polynomial synthetic division. Side-by-side, actually. Preceded by arithmetic long division (Greenleaf – an unfamiliar repeated subtraction without place value algorithm (how I was taught)), and the standard American algorithm. Opening examples were 279/13 and (2x^2 + 7x + 9)/(x + 3).

So I make them do a few examples, using both methods, synthetic and long, but before we can get to the 2007 game (thank you, Denise), a kid asks:

Which is better [long division or synthetic division]?

I froze, and then answered.

“In most high schools, polynomial long division is taught first, in an earlier course. Once synthetic division is taught, many teachers expect that it is the method to be used, all the time. But me, I never use synthetic division. Polynomial long division uses a familiar algorithm from arithmetic, and that is too big an advantage to ignore. In tonight’s homework, you have to do some of each, but for your own work, and on tests, you may choose which you are more comfortable with.”

Was I too honest?

36 Comments leave one →
1. February 13, 2007 am28 8:25 am 8:25 am

Of course not! I think students respect thoughtful honesty quite highly. Whenever there are multiple algorithms, there are usually advantages to each; a discussion is worthwhile.

2. February 13, 2007 am28 8:25 am 8:25 am

Wow, I thought I was the only one that didn’t really love synthetic division. I also find polynomial long division easier to keep track of, and prefer it. So it’s hard for me to be unbiased here but I generally think honesty is the best policy. If your students find synthetic division simpler and easier, they will do it even if you don’t. Of course, the way you demonstrate answers or give examples on the board will affect them strongly. What will you do from this point forward?

3. February 13, 2007 am28 10:42 am 10:42 am

I never liked synthetic division, either!

4. Clueless permalink
February 13, 2007 pm28 7:13 pm 7:13 pm

I agree with MRC, students will choose the method they like best if not forced to choose one by the teacher or the testing method.

I remember when I was in middle school, I came up with my own way of doing fraction addition, and was denied credit on a test because I did not follow the method prescribed (taught) by the teacher.

5. February 13, 2007 pm28 7:42 pm 7:42 pm

Sounds like common sense! In class, students need to do their best to understand the method being taught—especially since something may well build on it later. And in homework, if a certain method is specified in the instructions, students must practice that method. But in “real life” (which, in my opinion, includes a student taking a test), people can use whatever method they like, as long as it is a valid method. Polynomial long division is relatively easy to remember, because of the obvious parallel with regular long division. Synthetic division, while undoubtedly a shortcut…well, I’ll just say “ditto” to the posts above.

6. rdt permalink
February 13, 2007 pm28 8:37 pm 8:37 pm

My “real life” experience is that — unlike the quadratic formula, which I find I use relatively often — I’d completely forgotten both topics existed. So encouraging students to focus on the method that makes sense to them seems like a good idea to me, since its the method most likely to stick with them.

7. February 14, 2007 am28 3:52 am 3:52 am

sounds like honesty is still the best policy!
Here are some thoughts…
1) When my calc students needed to remember the quadratic factor in the difference of two cubes for some limit problem on a test, do you think any of them used long division? They were shown that synthetic is an abbreviated form and most adopt it and retain it for a while anyway. If htey forget hte method, they’ll go back to long division, nice to have an alternate!
2) When I choose to derive the derivative of x^n at x = ‘a’, without using the binomial formula, I quickly review patterns of factoring like x^3-a^3 but then use synthetic to obtain the quotient. Of course this could be done by long division, but I want their focus to be on the beautiful pattern of the resulting coefficients: a, a^2,a^3,…a^(n-1). Seeing these terms in just 2 rows is powerful stuff, but that may be my bias.
In the end, we need to come to terms with the amount of emphasis we should be placing on this topic. Outside of factoring and finding the roots of a polynomial equation without a calculator or software, can we make a strong case for this topic as much as in the past? Perhaps, since the division algorithm is a wonderful review of so many algebraic skills just as the arithmetic version is for arithmetic skills, but how much is enough? This is not obvious IMO, but a problem we should coming to terms with.
Of course, synthetic division isn’t practical when dealing with non-linear factors, so the div alg is needed in that case. Bigger issues regarding how much theory of equations should be included in the curriculum, like the rational root theorem, Descarte’s Rule of Signs (yup, some are still teaching that!), etc., need to be addressed. Members of major math curriculum committees like the NML believe strongly in the importance of the division algorithm and decry its current deemphasis. We need to thrash this out, see what other countries are doing, make a decision and get on with it…

8. February 14, 2007 am28 4:37 am 4:37 am

Teach whatever works. Give the kids a big toolbox, teach them how to choose appropriate tools, and encourage them to seek new tools and new applications.

I well remember our 7th and 8th grade math teacher, Roger Flick. He was the first real mathematician our school had ever hired. He encouraged lots of experimentation, and often provided extra reading from books featuring the old pros when he thought we might learn something from them. We’d scramble to find new ways to work problems, different from those offered by the book. Without exception, we’d find something cool, and Mr. Flick would provide us with a reference to something Euclid or Euler or some other hoary old math guy, showing that our great idea had been around a few hundred years.

We always thought it was cool that we’d discovered the idea independently. Most of the tricks I’ve not thought about in a long time, but I’m teaching geometry and algebra this week — they’re coming back quickly.

Tell you what, though, it’s difficult to slow down and not skip steps after a lifetime of not listing them all. Some of the tricks also defy recall. It’s great discovering this stuff again.

9. February 14, 2007 am28 5:12 am 5:12 am

I agree that kids should choose the method they like the best. I like synthetic division and my TI-89 calculator.

10. February 14, 2007 am28 9:04 am 9:04 am

Thanks for all the feedback.

MRC, I will continue to alternate methods on the board, as that stuff comes up. RDT is right, they don’t come up much (but of course today we built on them – remainder and factor theorems).

And thanks for all the variations on the advice to give them “a big toolbox” (nice phrase, Ed). There are only a few times I deny kids tools. When I am forming habits in algebra, I ban FOIL, I insist on transposition rather than pendant subtraction. The rest of the time it’s “the big toolbox.”

I feel better that a few of you (Denise, MRC, Darmok) agree with me (not my teaching) and find long division much more comfortable. Bigger thanks to Dave and Pissed Off Teacher for disagreeing… I’m only teaching 10 years, and not a big graphing calculator fan, and you know how I feel about long division. Both of you guys have what, double the years? and use the TI and synthetic. Makes a teacher think…

Clueless, how did you add fractions?

And for the record, yesterday three quarters of the kids favored long division. After using both methods in the homework, the class split about even, maybe leaning a bit synthetic.

Finally, I am a huge fan of Sarai’s blog, it carries an amazing sense of serenity. (math, fiddle, yarn). Thank you for braving the stress and mess of this place to read and leave a comment. Plus, in this case, as she both takes and teaches classes, I appreciate that ‘in-between’ perspective.

• July 26, 2012 pm31 5:16 pm 5:16 pm

What is “pendant subtraction”?

• July 26, 2012 pm31 5:19 pm 5:19 pm

Nevermind!

11. February 14, 2007 am28 9:56 am 9:56 am

I’m not sure if I should admit this, but I don’t think I was ever taught synthetic division, and if I was I do not have a slightest recollection of it! The long division always worked fine for me, but again, I never had to teach this in high school. As for calculators: I try to avoid them until I am certain that the students know why they are using them and how. In other words until I am certain that calculators are doing what students are able to do, just quicker.

12. February 14, 2007 am28 10:36 am 10:36 am

FOIL. First came across it today (remember, I’m a history/economics/geography guy) — acronym for something?

What’s wrong with it? I saw the name, but for the life of me I couldn’t find anything to distinguish what I saw from just solving for x.

13. Clueless permalink
February 14, 2007 pm28 6:34 pm 6:34 pm

Well, the teacher insisted that the common denominator be the LCM of the denominators of all fractions. I found it easier and faster to just use the product of the denominators, thus avoiding the calculation of the LCM.

14. February 14, 2007 pm28 8:40 pm 8:40 pm

I’ll post elsewhere about FOIL, but in short it’s an unnecessary mnemonic that cuts the feet out of lots of other ways of multiplying binomials that reinforce the distributive property or the connections to arithmetic mutliplication. I didn’t know you were a geography guy. I was a geography guy (once). I wonder if you ever used a map I edited…..

Clueless raises a common error. We should stop correcting this ‘mistake.’ We need common denominators, no need to fetishize the least of them. They are, intitially, just slightly more efficient. Eventually, with ugly polynomials, anything other than the least will be hard to work with, but that’s no reason to torture 3rd and 4th graders.

e, you teach?

15. eemina13 permalink
February 14, 2007 pm28 11:53 pm 11:53 pm

I do, but different population: college students. Currently the goal is to concentrate on preservice teachers :)

16. February 15, 2007 am28 12:08 am 12:08 am

sorry. that was e above.

17. February 16, 2007 am28 8:49 am 8:49 am

Was I too honest?

Not at all! I think that was a perfect answer.

18. Lsquared permalink
February 16, 2007 pm28 10:15 pm 10:15 pm

I like long division better too, but for a different reason–I’ve taught synthetic division out of too many books, and in some you change the sign and add, and in others you don’t change the sign and subtract (I think–though I could have that mixed up too…), so I can never remember what sign I’m supposed to be using for synthetic division until I either look it up again or figure it out using long division again.

Also, you said: ” I insist on transposition rather than pendant subtraction” which is language I don’t recognize. What are those?

19. February 17, 2007 am28 5:38 am 5:38 am

hmm, I made the phrase up … “pendant subtraction” …

y + 27 = 2x
y = 2x – 27 (that’s what I call transposition)

y + 27 = 2x
. -27 . . -27 (this is the line I ban)
y = 2x + 27

I probably need a full post to justify this.

20. February 18, 2007 pm28 10:53 pm 10:53 pm

I’m glad to see somebody does. Even worse is this:

y+27=2x -27
-27

Yikes.

21. June 18, 2007 pm30 7:14 pm 7:14 pm

Here’s an important fact about synthetic division:

If you teach it in high school, the probability is incredibly low that the students who learned it will remember it when they get to calculus in college. Because the long division algorithm is similar to long division for numbers, it’s much easier to remember.

Also, does anyone teach synthetic division where the divisor is quadratic or higher degree? The long division algorithm easily generalizes.

The students will need to know how to divide polynomials, but synthetic division is a waste of time.

As someone who teaches calculus every year, I recommend the following:

Teach long division.
Skip synthetic division.
Spend the extra time teaching the rational root test and the binomial theorem.

I always have to spend time in my calculus class teaching these things (usually before I do partial fractions) because almost none of the students know them.

22. February 12, 2008 am29 2:56 am 2:56 am

I’m very very late to this party, but just wanted to mention that synthetic division is all-but-unknown here in the UK. I had a maths PhD and several years of teaching experience before I heard of it for the first time. Long division (and dividing through linear factors “by inspection” where appropriate) are the only methods widely taught here.

23. February 22, 2008 am29 7:40 am 7:40 am

I too am late to this party, having arrived by looking for a way to get a TI-89 to symbolically represent the result of a polynomial by a binomial when the poly doesn’t factor into integer factors. So far I can only get it to factor out a common factor from the poly and show the result all over the binomial.
This conversation intrigued me because I am currently teaching this material in my algebra classes. I also show polynomial long division side by side with arithmetic long division, which special attention to the need for 0 place-holders. With regard to synthetic division, I find it a very easy and fast way to evaluate a higher order poly function at a given value. Making the connection between factoring out a linear factor and the result of the synthetic division being one degree less than the original polynomial provides another instructional opportunity. With regard to what sign to use, since it is based on division by (x-c), simply setting the divisor equal to zero and solving provides the correct divisor in the synthetic div., and also works with coefficients of x other than 1. (Helpful correlation with the rational zeros and related poly. curve sketching techniques in case you haven’t abandoned them for the fancy calculator)
Most of my work is with developmental level students in a community college, most of whom have been severely impaired by being presented with shortcuts and calculators. Explaining and insisting on work being done in full step by step detail consistently “turns on the light” for many who had been dismissed as hopelessly unable to do math.

24. February 22, 2008 am29 7:55 am 7:55 am

Too late? Impossible.

Some of the trick is how to win kids to avoiding unnecessary shortcuts.

Some is choosing, based on the class/level/students how much variety of technique to offer. And in some cases, how to choose between techniques.

I am leaning, these days, towards not teaching synthetic division for division (but neat for evaluating a function at a given value). Leaning, but I won’t be the only one around here not to teach it (next time I am teaching that level).

This topic continues…

25. Mrs. F permalink
July 23, 2008 am31 2:46 am 2:46 am

Even later to the discussion, but as I’m working through creating lessons for an Algebra class I keep coming across the question of efficiency (especially in preparing the toolbox of ways to solve a quadratic). The fact that your student asked the question means they are also thinking about which is most efficient or maybe not. I probably would have asked in response “What do you mean by better?”

Better = faster? Synthetic division wins.
Better = more memorable? Probably long division based on the fact that it is built on prior knowledge and is a longer process and can be remembered because of the labor and isn’t as much a thoughtless algorithm
Better = applicable in more situations? As was mentioned in a previous post, synthetic division is only used when the divisor is x – c. It can be done with a leading coefficient, but requires more steps and I don’t know of a way that it can be done with anything other than a linear factor.

As for you being too honest, I am frequently honest about the way I prefer things, but I don’t force my students to comply with my preferences unless it’s a matter of keeping mathematical integrity.

I don’t use FOIL either. It’s a 4-letter “f” word that is never mentioned in my classroom. It’s cute, but has limited applications! I’d rather not take up space in my students’ brain with it.

26. Mrs. F permalink
July 23, 2008 am31 2:47 am 2:47 am

PS: Anyone know who invented/discovered/formulated synthetic division? That’s how I came across this blog in the first place.

27. July 23, 2008 pm31 5:59 pm 5:59 pm

Wikipedia says it was described by Ruffino in 1809.

28. February 5, 2009 am28 1:23 am 1:23 am

Several comments mentioned synthetic division by a quadratic or higher polynomial power…
I cover both topics in two blogs
http://pballew.blogspot.com/2009/02/synthetic-divison-by-quadratic.html

You can divide synthetically by a polynomial of any power… As to the origin, my mind goes back to something in the Jiuzhang suanshu or Nine Chapters on the Mathematical Art but I am away from my notes and cannot confirm that…

29. February 5, 2009 am28 3:11 am 3:11 am

I skip synthetic division everytime it comes up.

30. July 4, 2009 pm31 3:00 pm 3:00 pm

jd2718,
Just last night I performed a side by side comparison between polynomial long division and a longer form of conventional long division. These can be exactly the same, differing only in the use of the factor, “10”, in the terms for dividend and divisor being replaced by a variable. Certain other basic arithmetic features need to be understood for the student or learner to best appreciate and finish evaluating the long division operation result. These other features are not overly difficult. I began dealing with methods to understand long division in my own way many many years ago. Really, the key to learning long division is to learn polynomial long division, unless one had already learned how to perform the algorithm correctly when still in early elementary school.

31. December 30, 2009 pm31 3:52 pm 3:52 pm

I used to skip synthetic, because I couldn’t remember how it went myself. But in one pre-calc class, the students wanted to learn it, so I offered extra credit if they’d teach it to each other. I learned it from them! :^) They were way happy to have a quicker way to test for factors when using the rational root theorem. Now, whenever I show polynomial division to my students (that level or higher), I mention that incident to them.

32. October 12, 2014 am31 6:45 am 6:45 am

experienced 1 or 2 problems with the survey shit, but at this point it payed back :))

33. Anonymous permalink
August 30, 2015 am31 11:46 am 11:46 am

i rather choose synthetic division than long division

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