Why Do I Teach Synthetic Division?

On Twitter today, Drew Lewis posed the following question:

I was going to try to answer but realized that 140 characters wasn’t enough and the tweetstorm I’d have to resort to was going to be too long. Fortunately it’s summertime, so I can actually sit down at the blog and explain why I do it.

First things first: I teach it in our Honor Precalculus course. If I were to teach a different course, I’d either do it differently than I’m about to describe, or I wouldn’t do it at all.

The reason I teach it is because it is a small piece of a big puzzle that prepares students for calculus, not from a content perspective (OK, a little from a content perspective) but from a keeping-track-of-all-the-details-in-a-big-complicated-problem perspective.

Here’s a pretty typical instruction from a Calculus textbook (Calculus, Anton, Bivens, and Davis, 10th edition, pg. 264): “Give a graph of the function and identify the locations of all critical and inflection points.” Here’s a pretty typical sequence of steps a student would take to answer this question:

  1. Take the derivative of the function.
  2. Set the derivative equal to zero and solve the resulting equation (keep in mind this could be a polynomial, rational, exponential, logarithmic, or transcendental function).
  3. Check the derivative for any points at which it is undefined.
  4. Take the second derivative of the function.
  5. Set the second derivative equal to zero and solve the resulting equation.
  6. Set up the appropriate tables or structures to identify intervals on which the function is increasing and decreasing, and justify the answer.
  7. Set up the appropriate tables or structures to identify intervals on which the function is concave up and concave down, and justify the answer.
  8. Find the y-intercept of the function.
  9. (Probably) Find the x-intercepts of the function.
  10. Sketch the graph of the function using the information gained from doing all this work.

Compare this to the following pretty typical instruction a student would encounter in a precalculus textbook: “Analyze the given polynomial function and sketch its resulting graph.” Here’s a pretty typical sequence of steps a student would take to answer this question:

  1. Factor the function if possible.
  2. If there are factors that are not linear or quadratic, ensure the function has integer coefficients or rewrite it so that it does.
  3. State all factors of the leading coefficient and constant terms, and use these factors to list all possible rational roots of the function.
  4. Test the possible rational roots until you find one that is a zero of the function.
  5. Divide the function by the appropriate factor to depress the degree of the polynomial.
  6. Continue steps 4 and 5 until you are left with linear and quadratic factors.
  7. Use the quadratic formula to obtain all linear factors over the set of real numbers.
  8. State all x-intercepts based on the factors you found.
  9. Find the y-intercept of the function.
  10. Determine the end behaviour of the function using the leading coefficient test.
  11. Sketch the graph of the function using the information gained from doing all this work.

As you can see, the process to answer each of these questions is quite long, requiring the use of definitions and theorems; recall of algebraic facts and algorithms; and the ability to keep the question asked in mind when the work involved is long and complicated. Are they conceptually equivalent? A little. Are they challenging for the students taking the respective courses? Yes. Does answering the precalculus question serve as a good foundation for answering the calculus question? I believe it does.

So why do I teach synthetic division? I teach it because it is a quick and easy algorithm to use in performing step 5 of the precalculus question, which is to say I’m giving them both one more tool in their bags and one more thing to understand and be able to apply. I think this helps prepare them for calculus. Otherwise, I can’t think of any reason why I would teach it (or most of the other stuff in that list).

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