Products Of Sums

The objective is an way to deal with a product of sums, particularly products of polynomials. Much of the difficulty lies in working with the large number of terms resulting from the initial expansion. The novelty here is that the expansion and the organizational part of post—processing steps are combined.

Method: "All Combinations"

This consists of a scanning procedure that gives terms in the expansion, and then decisions about what to do with them. This is a fairly elaborate organizational task and should be separated from arithmetic or algebraic manipulations.

Scanning, heuristic formulation: A product of sums can be written as a sum of products. The pieces are obtained by all possible ways of taking a term from each of the factors.

The key point is implicit: since we get all possible combinations the order in which they appear in the scanning procedure is an artifact and should not influence what we do with them.

Scanning, generic implementation: Designate one of the factors to be the “slow” factor. Step through the terms keeping track of the current one e.g. with a finger or by crossing out terms already done. For each such term scan the “fast” factor. The full expansion consists of all the corresponding products.

Example

Write as a polynomial in .

Detail of expansion

We collect terms by power of during the scanning process. A preliminary scan shows there will be terms in degrees 0 through 4. Set up places to record these:

(1)

Take the first first factor to be “slow”. Fix the first term and scan the second factor. Record products of coefficients in the proper places:

(2)

Note terms are recorded without any arithmetic or algebraic processing, even to the extent that parentheses are always used to avoid having to decide whether they are really needed. Trying to combine arithmetic and the organizational procedure frequently increases the error rate.

Proceed to the next term in the slow factor, combine with terms in the second and record:

(3)

The final term in the slow factor is done similarly.

The unprocessed outcome of the scanning process is

(4)

This could be used as a diagnostic aid. The final answer is obtained by doing the evident — and easy — processing of coefficients. Again we note that doing all algebra and arithmetic as a separate step significantly reduces errors, see Task Separation.