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When FOIL Fails

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


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,   

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   
Associated Topics:
High School Basic Algebra
High School Exponents
Middle School Algebra
Middle School Exponents

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