"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

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

Many instructors in later math classes come to hate "FOIL" because it serves mostly to confuse students when they reach more advanced topics. FOIL tends to be taught as "the" way to multiply all polynomials, which is clearly not true. (As soon as either one of the polynomials has more than a "first" and "last" term in its parentheses, you're hosed if you try to use FOIL, because those terms won't "fit".) When multiplying larger polynomials, just about everybody switches to vertical multiplication, because it's just so much easier to use.

If you want to use FOIL, that's fine, but (warning!) keep its restriction in mind: you can ONLY use it for the special case of multiplying two binomials. You can NOT use it at ANY other time!

• 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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 Cite this article as: Stapel, Elizabeth. "'FOIL': A Special (and Misleading) Case." Purplemath. Available from     http://www.purplemath.com/modules/polymult2.htm. Accessed [Date] [Month] 2016

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