Factorization

Date: 03/17/99 at 22:34:02
From: karen
Subject: Factorisation

Can you help me simplify 9b^2 - 24bc + 16c^2?

Date: 03/18/99 at 10:33:19
From: Doctor Rick
Subject: Re: Factorisation

Let us look at your goal. You want to get it in this form:

    2             2
  9b  - 24bc + 16c  = (Mb + Nc)(Pb + Qc)

You need to replace the 4 letters M, N, P, and Q with numbers so you 
will get the right coefficients.

The product MP must be 9, so M and P could be 1 and 9 or 3 and 3. 
(Order does not matter, since it is just a matter of which set of 
parentheses comes first.)

The product NQ must be 16, so N and Q could be 1 and 16, 2 and 8, or 
4 and 4. If M and P are 1 and 9 then order matters, so you would need 
to try 16 and 1, as well as 8 and 2.

The sum of the inside and outside products (MQ + NP) will be -24 
(remember the sign). Since it is negative, we must have at least one 
negative number, N or Q. But since NQ is positive, BOTH N and Q must 
be negative.

There are quite a few possibilities to try. But since both 9 and 16 
are perfect squares, my instinct says to try the squares first: 

   M = 3
   P = 3
   N = 4
   Q = 4 

Now make that N = -4, Q = -4 since they must be negative.

Try this out:

  (3b - 4c)(3b - 4c) = 9b^2 - 12bc             ... 3b(3b-4c)
                            - 12bc + 16c^2     ... -4c(2b-4c)
                       -------------------
                       9b^2 - 24bc + 16c^2

Sure enough, it works! Just for practice, let us see what would have 
happened if we tried something else first. What if M = 1, P = 9, and 
N = -8, Q = -2?

  MQ + NP = (1)(-2) + (-8)(9)
          = -2 - 72
          = -74

That is not close to -24, so we would have to try another combination. 
As you do enough of these, you will develop a feel for what might work 
and what will not. For instance, you will get a smaller (absolute 
value) bc term if you put the bigger numbers either both first or both 
last, so they do not get multiplied together in the cross term (bc). 
With M = 1, P = 9, and N = -2, Q = -8:

  MQ + NP = (1)(-8) + (-2)(9)
          = -8 - 18
          = -26

That is a lot closer to what we wanted (-24). My reasoning turned out 
to be correct. Of course, we already have the answer, and it is not 
very close to what I just tried. There is a lot of trial and error in 
factoring, but as I said, you can develop a feel for it.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/