Sunday, September 16, 2012

Generic Rectangles

Quite a while ago I posted about using the grid method for dividing polynomials. Using grids or generic rectangles as they are more commonly called, makes dividing polynomials seem like doing a crossword puzzle or a sudoku.

There are number of ways to use generic rectangles, and I thought I would try to do a few posts about some of their other uses, which include factoring trinomials and completing the square. This first post is just going to be about polynomial multiplication.

When multiplying, generic rectangles seem to provide for polynomials what grid multiplication and lattice multiplication provide us for numbers - a way of arranging things on the page so that we don't mess up our calculations. Grid and lattice multiplication allow us to break down our numbers into more manageable chunks and, in the case of lattice multiplication, give a way to keep track of "carries." In school, grid and lattice multiplication are sometimes presented as an alternative to standard "long" multiplication.

Those familiar with grid and lattice multiplication for numbers will feel at home with generic rectangles, but there are some differences. In the world of polynomial multiplication, there are no "carries" (terms in a polynomial, unlike decimal places don't overflow into each other), and there is no generally used "long multiplication" style that needs replacing. For polynomials, what generic rectangles give us is a way to keep track of all those terms that come out of the distributive law, which most students struggle to keep track of using mnemonics like FOIL. (Once recursion becomes something well-understood in middle school,  FOIL might be a reasonable way to teach the distributive law, but until then, teachers please consider using generic rectangles.)

Generic rectangles also have an affinity with algebra tiles - a manipulative that is sometimes for learning polynomial multiplication. If you use algebra tiles, generic rectangles are a nice thing to move on to if you tire of pushing around all that plastic. Unlike those rigid tiles, generic rectangles are more generic: although they don't provide the full force of the area metaphor for multiplying that algebra tiles do they allow you to multiply any kind of terms.

A first example
The first thing to do when provided with two polynomials that are to be multiplied is to set up the grid, with the terms from one of the polynomials across the top row, and the terms from the other down the leftmost column.

Each term is multiplied, like terms are gathered, and the results are summed, which provides the answer.

A little bigger example
Here's an example with a trinomial and a binomial.

Again, step 1 is just putting the factors to be multiplied on the left most and topmost column and row of the grid, and step 2 is just completing the term-by-term multiplication to fill in the grid.

Finally, like-terms are collected and the final result can be written out.

An infinite example
That's fun, but what about multiplying together two infinitely long polynomials? Why not? Of course, we'll  quickly run out of space, but we might see something in our grid that will help us get to a solution.  Now, instead of thinking of polynomials in the way you might usually think of them, you should consider these infinitely long polynomials as formal power series - polynomials that we will never evaluate, and that we are just treating as worthwhile mathematical objects in their own right, regardless of whether they ever converge.

Consider the product:

We can make a grid like this:

And start to fill it in:

Do you see the pattern that is emerging when you start collecting like terms? Once you do you'll probably agree that:
If you go a little further with this, you'll find the triangular numbers lurking in here. For  d > 0:
The power series on the right has coefficients that are the "d-1"-dimensional triangular numbers (or, the d-1 column of Pascal's Triangle - see this post).