Date: Jul 29, 2007 2:31 PM
Author: Stephen J. Herschkorn
Subject: AC method of factoring polynomials

Summary:  How well known and/or frequently taught is the AC method of 
factoring, sometimes called factoring by grouping.

Factoring of polynomials often seemed like an art to me. For example,

18x^2 + 7x - 30.

I used to consider all possible pairs of factors of 18 and of 30 until
I found the right coefficients. Considering placement of thes factors,
that's 24 possible combinations, though with intuition (hence the art),
I might be able to narrow down the search.

From a current client's textbook on College Algebra, I only recently
learned a method the book calls "factoring by grouping." The client's
professor calls it the "AC method," from consideration of polynomials of
the type Ax^2 + Bx + C. Here's how it works in the above example:

- Multiply the leading coeffiecient 18 = 2 x 3^2 and the constant
term -30 = -2 x 3 x 5, getting -540 = -2^2 x 3^3 x 5.

- Find a pair of factors of -540 such that their sum is the middle
coefficient 7. That is equivalent to findiing factors of 540 whose
difference is 7. Either by listing all the factors or by looking at the
prime factorization, we find 20 = 2^2 x 5 and 27 = 3^3 as these
factors. I prefer the prime factorization way, in which case I didn't
even need the fact that the product was 540.

- Rewrite the polynomial by splitting up the middle term: 18x^2 +27x -
20x + 30. (-20x + 27x will work as well.)

- Factor by grouping: 9x(2x + 3) - 10(2x + 3) = (9x -10) (2x + 3).
Voil`a! (grave accent)

- If no pair of factors of AC (the product of the leading coefficient
and the constant term) sum to the middle coefficient B, then the
polynomial is irreducible.

When A > 1, this approach seems in general a lot easier to me than
searching pairs of factors of A and C individually. If you haven't
seen this before, try it on some examples yourself, such as

6x^2 + 13x y + 6y^2
16a^4 - 24a^2 b + 9b^2
12x^2 - 29x + 15
6b^2 + 13b - 28
10m^2 -13m n - 3n^2

I don't think it is the case that I learned this method once long ago
and subsequently forgot it, so I am surprised I never saw it before.
How well known is this method? Is it taught much? I don't find it in
my favorite College Algebra text (by C.H. Lehmann), and it doesn't show
up in the first three pages from Googl(R)ing "polynomial factor." At
least one of my more advanced clients had never seen it before either.

Stephen J. Herschkorn
Math Tutor on the Internet and in Central New Jersey and Manhhattan