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This page explains FOIL, which means First, Outer, Inner, Last.

It's the method for multiplying two expressions, each of which has two terms.

How to recognize a F.O.I.L. situation

(A + B) (C + D)
 EXAMPLE (3x + 8) (2x - 3)

Do you see that A, B, C, and D can contain numbers and symbols?  Also, do you see that in our example, D represented -3, a negative number?

First, I'll remind you of the distributive property of addition over multiplication.  It states:

The sum of A and B, multiplied by any number or expression E is equal to the sum of A multiplied by E and B multiplied by E.  In symbols, it can be stated as follows:

(A + B) (E)  =  AE + BE

 EXAMPLE (3x + 8) (w) can be expressed as follows: 3xw + 8w

Now, if E = C + D, we can substitute C + D in place of E in this equation, as follows:

(A + B) (C + D) = A (C + D) + B (C + D)

Now, the distributive property can be used two more times as follows:

(A + B) (C + D) = AC + AD + BC + BD

Notice that AC (the product of A and C) are the first terms, AD are outer terms, BC are inner terms, and BD are outer terms -- first, outer, inner, and last, or F.O.I.L.

Now, let's return to our first example:

 EXAMPLE (3x + 8) (2x - 3)

The first terms are 3x and 2x.  Their product is 6x².

The outer terms are 3x and -3.  Their product is -9x.

The inner terms are 8 and 2x.  Their product is 16x.

The last terms are 8 and -3.  Their product is -24.

Adding them up, the sum is 6x² - 9x + 16x - 24.

Simplifying, the sum is 6x² + 7x - 24.

If you're smart, you have already figured out that simplifying using F.O.I.L. is the opposite of factoring.  In other words, if you are asked to factor the trinomial 6x² + 7x - 24, you will reply (3x + 8) (2x - 3).  If you are asked to simplify (3x + 8) (2x - 3) using F.O.I.L., you will reply 6x² + 7x - 24.

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The webmaster and author of this Math Help site is Graeme McRae.