Why Factor?

Date: 08/18/2001 at 16:54:19
From: Kevin O'Neil
Subject: Factoring in algebra

I haven't seen this answer yet anywhere I've looked. WHY specifically 
does one have to factor out a problem? My daughter is homeschooling 
and we have an excellent Algebra 1 teacher, but it seems to be a 
hard question to answer. I've been explained the how's, the where's, 
the what's, but not the WHY. My daughter seems to need a practical 
explanation in order to understand math or how it applies to 
something. If you had a problem like  4b^2y^2 - 8b^2y + 24b^2 =  ? 

What is the practical use of factoring this out? I know it's like 
unmultiplying, but trying to explain that without a reason "why" 
behind it gets my daughter confused. She wants to know where it is 
used and why it is used, to help her figure out how. What applications 
would it serve? An example where a formula like that might be used 
would be helpful. Why not just solve the problem anyway? Aren't all 
the individual pieces needed there to solve it?

Date: 08/20/2001 at 18:01:07
From: Doctor Ian
Subject: Re: Factoring in algebra

Hi Kevin,

If I understand you correctly, you want to know why the expression 

  4b^2(y^2 - 5y + 6) =  ? 

is preferable to the expression

  4b^2y^2 - 20b^2y + 24b^2 =  ? 

Well, the main reason for identifying common factors is to let you see 
more deeply into the pattern. In this case, once we get the 4b^2 out 
in front, we can see that we have a standard quadratic form in y, 
which we can simplify even more:

  4b^2y^2 - 20b^2y + 24b^2 =  4b^2(y^2 - 5y + 6)

                           = 4b^2(y - 2)(y - 3)

Now, why is _this_ form preferable?  Well, for one thing, if the '?' 
is a zero, as it is in 'standard' form, then we know that

  4b^2(y - 2)(y - 3) = 0

Just by looking at this, we can see that if b is non-zero, there are 
only two possible values of y that can make this equation true:  
y = 2 and y = 3.  

It's also easy to see that halfway in between those two values, 

  4b^2(2.5 - 2)(2.5 - 3) = 4b^2(0.5)(-0.5)

the value of the expression is negative, which means that since we 
know that every quadratic expression is the description of a parabola, 
we now have enough information to draw a pretty good graph of the 
expression. Try doing that with the original form of the equation. 

Note that this also tells us that if we vary the value of b, we can 
alter the shape of the parabola, but we can't move the axis of 
symmetry (since that would require us to change where it hits the 
x-axis).  

In general, when mathematicians develop a technique like factoring out
common terms, you can be sure it's because they've found that it turns 
some really tedious and error-prone task (like trying to draw the 
graph of a function) into a very easy one, e.g., 

   Why study Prime and Composite Numbers?
   http://mathforum.org/library/drmath/view/57182.html   

Have you heard the joke in our archives toward the end of this answer?

   Quadratic Formula: Solve for b
   http://mathforum.org/library/drmath/view/52370.html   

Such jokes expose a deep truth about the way mathematicians do 
mathematics, and I mention them because one way to make learning math 
more entertaining is to try to approach the subject the way a 
mathematician would.  

I'm not saying that you should learn to enjoy jotting down pages and 
pages of equations!  What I mean is that you should get into the habit 
of looking for the easiest possible way to get from the statement of a 
problem to its solution, which in many cases means finding a short 
connection between the problem you're working on and one that you know 
has already been solved... whether or not you were the one who solved 
it.  

Factoring things out is, ultimately, a technique for helping you do 
this. 

As you've noted, you can solve problems without factoring out common
terms... and you can get from New York to Los Angeles on foot, if you 
want to, but it will take longer.  

I hope this helps.  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

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