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 ("outer" terms multiplied)
Inner ("inner" terms multiplied)
Last ("last" terms multiplied)

The answer is then the sum of the terms obtained. Thus the general form is

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

[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.