Re: AC method of factoring polynomials
- From: tommy1729 <tommy1729@xxxxxxxxx>
- Date: Sun, 29 Jul 2007 17:41:35 EDT
quasi wrote :
On Sun, 29 Jul 2007 14:31:47 -0400, "Stephen J.
Herschkorn"
<sjherschko@xxxxxxxxxxxx> wrote:
Summary: How well known and/or frequently taught isthe AC method of
factoring, sometimes called factoring by grouping.me. For example,
Factoring of polynomials often seemed like an art to
consider18 and of 30 until
18x^2 + 7x - 30.
I used to consider all possible pairs of factors of
I found the right coefficients. Consideringplacement of thes factors,
that's 24 possible combinations, though withintuition (hence the art),
I might be able to narrow down the search.Algebra, I only recently
From a current client's textbook on College
learned a method the book calls "factoring bygrouping." The client's
professor calls it the "AC method," fromconsideration of polynomials of
the type Ax^2 + Bx + C. Here's how it works in theabove example:
and the constant
- Multiply the leading coeffiecient 18 = 2 x 3^2
term -30 = -2 x 3 x 5, getting -540 = -2^2 x 3^3 x5.
is the middle
- Find a pair of factors of -540 such that their sum
coefficient 7. That is equivalent to findiingfactors of 540 whose
difference is 7. Either by listing all the factorsor 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, inwhich case I didn't
even need the fact that the product was 540.term: 18x^2 +27x -
- Rewrite the polynomial by splitting up the middle
20x + 30. (-20x + 27x will work as well.)(9x -10) (2x + 3).
- Factor by grouping: 9x(2x + 3) - 10(2x + 3) =
Voil`a! (grave accent)leading coefficient
- If no pair of factors of AC (the product of the
and the constant term) sum to the middle coefficientB, then the
polynomial is irreducible.easier to me than
When A > 1, this approach seems in general a lot
searching pairs of factors of A and Cindividually. If you haven't
seen this before, try it on some examples yourself,such as
method once long ago
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
and subsequently forgot it, so I am surprised Inever 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 seenit before either.
It's sometimes called "the master product" method.
In my opinion, it _shouldn't_ be taught -- it's too
artificial.
Sure it works, and it's not hard (for us) to prove
that it works, but
for the student of elementary algebra, it's just some
kind of
meaningless magic.
Moreover, it obscures the much simpler, much more
basic idea that the
leading coefficients of the factors must produce the
leading
coefficient of the product, and similarly for the
constant terms.
Also, students are not in the _business_ of factoring
non-monic
quadratics. It's not like there are going to be that
many of them.
Much better if they simply work through factor tables
for both the
leading and trailing coefficients, since that method
that actually
makes sense, even if it's a little longer.
Besides, at the next level of algebra (intermediate
algebra or
precalculus), you can factor a non-monic quadratic by
first completing
the square -- thus, no trial and error at all.
Alternatively, you can
use the quadratic formula to find the roots, and from
the roots, you
can deduce the factors.
The students have enough to learn. No need to burden
them with obscure
methods that get obsoleted at the next level.
quasi
i dont quite agree.
i agree on not needing to burden them , and not making a big deal about it in the exams ...
but for the students who want to know as much as possible , why keep that behind ??
i know ; i like factoring.
and perhaps it is better to teach this at number theory rather than algebra.
it should not effect the grades very much though , since it is not that important.
and the method is a bit controversial ....
i like the fact of irreducable 'detector' better than its purpose actually
tommy1729
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