This page explains FOIL, which means First, Outer, Inner, Last.
method for multiplying two expressions, each of which has two terms.
|How to recognize a F.O.I.L. situation
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
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:
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.
Introduction to factoring a quadratic expression
The webmaster and author of this Math Help site is Graeme McRae.