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/
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