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,

consider

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 sjherschko@netscape.net

Math Tutor on the Internet and in Central New Jersey and Manhhattan