When FOIL Fails

Date: 03/22/2001 at 22:32:27
From: Ryan Sarabia
Subject: FOIL fails

Hello,

My name is Ryan and I am in 8th grade. My question is:

Find the product - (2c-3)3rd power

I have examples in my book about using the FOIL process.
F - First
O - Outer
I - Inner
L - Last

I got as far as (2c-3)(2c-3)(2c-3).

I can do FOIL on a question to the second power, but I cannot figure 
out how to do a question to the third power. Please help.

Thank You.

Date: 03/23/2001 at 11:16:23
From: Doctor Ian
Subject: Re: FOIL fails

Hi Ryan,

This is a perfect example of why depending on 'tricks' like FOIL can 
do more harm than good. You don't need FOIL if you understand the 
distributive property:

  a(b - c) = ab - ac 

Let's look at a slightly different problem: (2 - k)^3. As you've 
noted, we can expand this to

  (2 - k)(2 - k)(2 - k)

Now we can apply the distributive property:

        a        b   c          a        b          a        c
  |------------|          |------------|      |------------|
  (2 - k)(2 - k)(2 - k) = (2 - k)(2 - k)(2) - (2 - k)(2 - k)(k)


                        = 2(2 - k)(2 - k) - k(2 - k)(2 - k)

Now we can do the same thing again:

                             a     b   c       a     b   c
                          |------|          |------| 
                        = 2(2 - k)(2 - k) - k(2 - k)(2 - k)


                            a    b      a   c     a   b     a   c
                          |----|      |----|    |----|    |----|
                        = 2(2-k)(2) - 2(2-k)k - k(2-k)2 + k(2-k)k


                        = 2^2(2-k) - 2k(2-k) - 2k(2-k) + k^2(2-k)                     

If you do it one more time, you should have a complete expansion. 

Note that FOIL has exactly the same effect as applying the 
distributive property twice:

  (a + b)(c + d) = (a + b)c + (a + b)d

                 = ac + bc + ad + bd
                  
                 = ac + ad + bc + bd    (first, outside, inside, last)

If you just memorize FOIL without understanding _why_ it works, you'll 
be stuck when it doesn't. But if you understand _why_ it works, then 
you don't have to bother memorizing it. 

Note that there is another trick, called Pascal's Triangle,
  
  http://mathforum.org/dr.math/faq/faq.pascal.triangle.html   

that you can use to solve problems like this quickly, but the same 
warning applies - if you learn _why_ the tricks work, you don't have 
to memorize the tricks.  

In fact, if you use the distributive property to work out a few 
expansions like (a + b)^4, (a + b)^5, and so on, you'll start to 
discover some patterns for yourself, and those will stay with you a 
lot longer than if someone else tells you about them. 

The moral of the story is: When someone tries to teach you a trick 
(like FOIL), make sure that you can explain _why_ it works.  If you 
can't, don't use it. 

I hope this helps. Let me know 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/