Multiplication & Division of Polynomials

When I was in high school I learned to multiply polynomials by using a technique that was reminiscent of the multiplication of multi-digit numbers that I had learned in earlier grades. That is, after filling in any "missing terms" with zeros the two polynomials were written one over the other, with the terms arranged by degree, with the constant term at the right. If the polynomials were of degrees m and n, with m <= n, the polynomial of degree m was written on the bottom. We would multiply the top polynomial by each of the m terms of the bottom polynomial, to produce m rows, each of which was indented one position to the right. Then all terms of the same degree were located above one another, and we would add each column to get the answer. Similarly, the algorithm for polynomial long division was based on the algorithm for multi-digit integer division. Polynomial division never really clicked for me. I learned how to do it, but somehow it felt like magic.

These days, most elementary and middle school students - even many students from better schools - are not being taught multiplication and division of multi-digit numbers, instead being told they can rely on the calculator app on their smartphone. This has caused much hand-wringing, particularly among traditionalists. How can we teach students to multiply and divide polynomials when they haven't learned to multiply and divide integers? Non-traditionalists might say that students can use CAS to multiply and divide polynomials and don't need to be able to do it by hand, but I am not willing to go that far.

In the course of tutoring a high school student, I recently was exposed to a method for multiplying polynomials that is being taught currently in some schools, called the galley method . I really like it, for several reasons:

(1) The method does not not depend on knowing the analogous arithmetical algorithm.
(2) The method is directly related to the definition of the multiplication of m and n as the area of an m x n rectangle (or as the cardinality of the Cartesian product of sets of cardinalities m and n, if you prefer). These definitions are, in the opinion of many educators, superior to regarding multiplication as repeated addition.
(3) The first time I used the galley method, I found it slightly easier to use than the traditional method. I would use it myself in cases where a CAS is not necessary or available.
(4) The galley method immediately leads to the reverse galley method for the division of polynomials.

Here's how it works: To multiply polynomials of degree m and n, create an (m+1) by (n+1) table, the galley. (The term galley apparently comes a French word that indicates an oblong tray for holding setup type.) Write the terms of the degree n polynomial above the n+1 columns, in order of decreasing degree as you move from left to right. Write the terms of the other polynomial along the right side of the galley, in order of decreasing degree. Then, fill in each cell with the product of the monomials above its column and to the right of its row. Finally, observe that the terms on any diagonal (of slope 1) have the same degree, and the degrees corresponding to different diagonals are different. Add along these diagonals to combine like terms, and write the result in the first empty space outside the galley. The result is now in front of you, in an L-shaped pattern along the left and bottom sides of the galley. If you find this difficult to visualize, you can find an excellent description with diagrams here.

The galley method can be used in reverse for polynomial division. Suppose you want to divide a polynomial of degree m by a polynomial of degree n, where n < m. Create a galley of n+1 rows and m-n+1 columns. Write the n+1 terms of the divisor along the right side of the galley and the m+1 terms of the dividend in an L-shaped band beginning in the area one unit to the left and one unit below the upper left-hand corner of the galley. Then fill in the terms in the remainder of the galley as well as the terms above the galley one by one, beginning with row 1 column 1 of the galley, then row 0 column 1 (just above row 1 column 1), then the rest of column 1. Next move to row 1 column 2, and so forth. The solution will appear above the galley. If there is a remainder, some of the last n-1 terms in the divisor (the L-shaped band) will not in fact be the sum of the terms on their diagonals, and the remainder is the difference between the sum of the terms on the diagonals and the corresponding terms in the divisor. Again, if you need further description, go here.