Concept of Factoring

Date: 03/22/97 at 13:22:54
From: James Torres
Subject: Concept of Factoring

My son is having problems with "factoring". I would like to know 
what he is talking about. Could you explain the concept of factoring 
and give examples of its practical uses?

Thank you for your help,

Jim Torres,

Date: 03/26/97 at 23:52:34
From: Doctor Jodi
Subject: Re: Concept of Factoring

Hi Jim,

Thanks for your question. Factoring is an idea you might be somewhat 
familiar with from multiplication. The numbers multiplied together to 
get another number are its factors.  

For example, 4*3 = 12, so 3 and 4 are factors of 12.  However, they're 
not its ONLY factors. 1, 2, 6, and 12 are other factors of 12.  
(Another way of defining a factor is a number which goes evenly into 
the number you're factoring.)

The sort of factoring your son is doing is somewhat similar, but since
it's algebra, there are probably LETTERS like x and y stuck into the 
equations. These letters just stand for UNKNOWNS.

For example, you could have an equation like x + 1 = 5.  You can ask
yourself, what number, added to one, gives you five? X stands for that
number.

You could figure out the equation x + 1 = 5 by subtracting 1 from each
side:

   x + 1 = 5
      -1   -1
   -----------
       x = 4

Other algebraic equations (equations with letters) might look 
different, but the idea behind them is the same.

So what does this have to do with factoring? Sometimes you'll have 
an equation that has a squared or cubed term which is unknown. 
Here's an example:  x^2 = 9

Since the x is squared, there are TWO possible answers for x. 
You can probably guess that the answers will be -3 and 3, since
-3 * -3 = 9 and 3 * 3 = 9.

But suppose you don't know what the answer is (or suppose that you're 
dealing with a more complicated equation). How would you figure out 
what the answer is?

Here's where we get back to factoring. Here's what you'd do:

   x^2 = 9
    -9   -9
   ---------
   x^2 - 9 = 0

Now, here's the trick. Remember, if you multiply 5 * 0, you get 0. In 
fact, if you multiply 28, or 1 billion, or any other number by 0, you 
get 0. And if you multiply and get 0 as an answer, at least one of the 
numbers you multiplied by MUST HAVE BEEN ZERO!

We can use that fact here.  We know that x^2 - 9 = 0.  So one of the 
factors of x^2 - 9 must equal zero. Does that make sense?

Next, we factor. Basically, this means saying to ourselves:

      What do we multiply to get x^2 - 9? 

There are a lot of tricks here, and I'm sure your son would be happy 
to show you some of them. 

From x^2 - 9 = 0, we get (x-3)(x+3) = 0.

You could multiply this out and check it, but since factoring itself 
is the issue here, I won't. If you want help on that, ask your son, or 
write back to us.

What it boils down to is breaking the equation into two parts (one
for each of the two possibilities). Either x-3 could be zero or x + 3 
could be zero.

If x - 3 =  0, then 
     + 3   + 3
     ------------
       x = 3. 

If x + 3 =  0, then 
     - 3   - 3
     ------------
       x = - 3. 

I hope this isn't too overwhelming. I applaud your interest in the 
math your son is doing. Sometimes it's hard to stay involved in 
teenagers' life at school. Even though they may shrug when you ask 
what they did at school, or complain that you're too nosy, I think 
most teenagers will be glad that you take an interest in what they're 
doing. 

Too often, kids feel limited by the knowledge (or lack of knowledge) 
of their parents. Learning as much as you can--and being curious about 
their lives at school--really shows them how important learning is.  
Going out of your way to learn things that might be difficult at first 
also shows them how important they are to you.

-Doctor Jodi,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/