FOIL rule

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The FOIL rule, also sometimes known as the double distributive property or more colloquially as foiling, is commonly taught to students learning algebra as a mnemonic for remembering how to multiply two binomials (polynomials with two terms). The name comes from the order of multiplying terms of the binomials:

First ("first" terms of each binomial are multiplied together)
Outer ("outside" terms multiplied)
Inner ("inside" terms multiplied)
Last ("last" terms multiplied)

FOIL was also interpreted as FOIB, standing for:

First
Outer
Inner
Back

Image:FOIL method.jpg

FOIL_method.JPG‎ (372 × 328 pixel, file size: 17 KB, MIME type: image/jpeg) The answer is then the sum of the terms obtained. Thus the general form is

$(a+b)(c+d) = \underbrace{ac}_\mathrm{first} + \underbrace{ad}_\mathrm{outside} + \underbrace{bc}_\mathrm{inside} + \underbrace{bd}_\mathrm{last}.\,$

1 Examples
2 Proof
3 Teaching
- 3.1 The FOIL Dance
4 Factorization

[edit] Examples

$(x + 2 y)(x + 1) = x 2 + x + 2 x y + 2 y$
$(x + 5)(x + 7) = x 2 + 7 x + 5 x + 35 = x 2 + 12 x + 35$
$(x + 1)(x - 1) = x 2 - x + x - 1 = x 2 - 1$

[edit] Proof

The FOIL rule can be shown to be equivalent to two applications of the distributive property.

(a + b)(c + d)

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

= a c + a d + b c + b d

[edit] Teaching

The FOIL mnemonic is commonly taught but is sometimes frowned upon because the method does not work, without modification, for higher order polynomials (the double distributive method, by contrast, is easily extended to the latter case). Foiling can thus be seen as an example of learning by rote memorization of rules rather than by understanding underlying concepts.

[edit] The FOIL Dance

The FOIL Dance is a dance taught by some Algebra teachers and, though uncommon, is a good dance to help students learn what to do when multiplying two binomials. It is a four-part dance, it's very simple to learn. You start out standing up, feet about shoulder width apart. Move your thumbs towards the left of your body, do the same thing with your feet and hips. This resembles the "Firsts". Now shift your left thumb facing the left side of your body, and your right thumb to the right side. Point your feet directly away from each other, and bend your knees in the same manner. THis resembles the "Outers". Point your toes towards the inside of your body, and do the same thing with your thumbs. This is the "Inners" "Lasts" is the same thing as "Firsts", except to the right instead of to the left.