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"FOIL": A Special (and Misleading) Case (page 2 of 3)

Sections: Simple multiplication, "FOIL" (and a warning), General multiplication

There is also a special method, useful ONLY for a two-term polynomial times another two-term polynomial. The method is called "FOIL". The letters F-O-I-L come from the words "first", "outer", "inner", "last", and are a memory device for helping you remember how to multiply horizontally, without having to write out the distribution like I did, and without dropping any terms. Here is what FOIL stands for:

That is, FOIL tells you to multiply the first terms in each of the parentheses, then multiply the two terms that are on the "outside" (furthest from each other), then the two terms that are on the "inside" (closest to each other), and then the last terms in each of the parentheses. In other words, using the previous example:

• Use FOIL to simplify (x + 3)(x + 2)

"first":  (x)(x) = x²
"outer":  (x)(2) = 2x
"inner":  (3)(x) = 3x
"last":  (3)(2) = 6

So:   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

(x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6

• Simplify (x – 4)(x – 3)

So the answer is: x² – 7x + 12

Using FOIL would give:

"first": (x)(x) = x²
"outer": (x)(–3) = –3x
"inner": (–4)(x) = –4x
"last": (–4)(–3) = +12

product: (x²) + (–3x) + (–4x) + (+12) = x² – 7x + 12

• Simplify (x – 3y)(x + y)

So the answer is: x² – 2xy – 3y²

Using FOIL would give:

"first": (x)(x) = x²
"outer": (x)(y) = xy
"inner": (–3y)(x) = –3xy
"last": (–3y)(y) = –3y²

product: (x²) + (xy) + (–3xy) + (–3y²) = x² – 2xy – 3y²

Let me reiterate what I said at the beginning: "FOIL" works only for the specific and special case of a two-term expression times another two-term expression. It does not apply in any other case.

You should not rely on FOIL for general multiplication, and should not expect it to "work" for every multiplication, or even for most multiplications. If you only learn FOIL, you will not have learned all you need to know, and this will cause you problems later on down the road.

I have seen too many students greatly hampered in their studies by an unthinking over-reliance on FOIL. Their instructors often never even taught them any method for multiplying other sorts of polynomials. For your own sake, take the time to read the next page and learn how to multiply polynomials properly.

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