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The FOIL Method

Author: Sophia
PDF Version

what's covered
In this lesson, you will learn how to multiply two binomials by using the FOIL method. Specifically, this lesson will cover:

Table of Contents

1. The Distributive Rule

Before we introduce the FOIL method, it is helpful to review the distributive rule, because the two are similar processes. The distributive rule is used when a quantity is being multiplied by a sum.

formula to know
Distributive Property
a open parentheses b plus c close parentheses equals a b plus a c

EXAMPLE

3 left parenthesis 5 plus 9 right parenthesis can be evaluated using distribution.

3 left parenthesis 5 plus 9 right parenthesis The 3 on the outside is multiplied into each term in the parentheses
open parentheses 3 times 5 close parentheses plus open parentheses 3 times 9 close parentheses Find each sum
15 plus 27 Evaluate
42 Our Solution

We can confirm this by evaluating 3 left parenthesis 5 plus 9 right parenthesis using the order of operations: 3 left parenthesis 5 plus 9 right parenthesis equals 3 left parenthesis 14 right parenthesis equals 42.

2. The FOIL Method

Earlier, we saw how distribution can help us evaluate expressions in the form a open parentheses b plus c close parentheses, by distributing a into b plus c to get a b plus a c. How could we distribute something like open parentheses 4 plus 3 close parentheses open parentheses 5 plus 1 close parentheses?

We can evaluate such expressions through distribution, but the process works in a slightly different way. What really happens is that we distribute twice: first, we distribute 4 into 5 and 1, then we distribute 3 into 5 and 1. Take a look at how this distribution works:

EXAMPLE

open parentheses 4 plus 3 close parentheses open parentheses 5 plus 1 close parentheses Distribute 4 into open parentheses 5 plus 1 close parentheses
4 times 5 plus 4 times 1 Distribute 3 into open parentheses 5 plus 1 close parentheses
3 times 5 plus 3 times 1 Evaluate and combine all parts
20 plus 4 plus 15 plus 3 Add
42 Our Solution

Taking a look at the distributions, we can say that in our first step, we multiplied the first terms of each factor. 4 is the first term in (4 + 3) and 5 is the first term in (5 + 1), to get 20. Our next step was to multiply the two outer terms. 4 and 1 are the outermost terms in our expression, and when multiplied, we get 4. In the next step, we multiplied the two inner terms, 3 and 5, to get 15. Finally, we multiplied the last terms in each factor: 3 is the last term in (4 + 3) and 1 is the last term in (5 + 1). When multiplied together, this equaled 3. Put them all together and we got 20 plus 4 plus 15 plus 3 equals 42.

In short, we multiplied the first term in each factor, then the outside two terms, then the inside two terms, and then the last term in each factor. This is known as the FOIL method: First, Outside, Inside, Last.

hint
Using FOIL to evaluate numerical expressions may seem odd, but FOIL is an extremely useful method when working with quadratics and algebraic expressions. Practicing without variables will help us see the properties and relationship to distribution, which will make FOILing algebraic expressions much easier!

term to know
FOIL
An acronym to remember the steps for distributing factors in binomial multiplication: first, outside, inside, last.

3. Practice Using FOIL

Let's take a look at some more examples of using FOIL to evaluate expressions. As we work through these examples, pay attention to the sign of the numbers. We bring positive and negatives with us when distributing!

EXAMPLE

open parentheses 5 minus 2 close parentheses open parentheses 4 plus 7 close parentheses Distribute 5 into open parentheses 4 plus 7 close parentheses
First colon thin space 5 times 4 comma space Outside colon thin space 5 times 7 Distribute -2 into open parentheses 4 plus 7 close parentheses
Inside colon thin space short dash 2 times 4 comma space Last colon thin space short dash 2 times 7 Combine all parts
20 plus 35 plus open parentheses short dash 8 close parentheses plus open parentheses short dash 14 close parentheses Change to subtraction
20 plus 35 minus 8 minus 14 Evaluate
33 Our Solution

EXAMPLE

open parentheses 6 minus 3 close parentheses open parentheses 8 minus 5 close parentheses Distribute 6 into open parentheses 8 minus 5 close parentheses
First colon thin space 6 times 8 comma space Outside colon space 6 times short dash 5 Distribute -3 into open parentheses 8 minus 5 close parentheses
Inside colon thin space short dash 3 times 8 comma space Last colon thin space short dash 3 times short dash 5 Combine all parts
48 plus open parentheses short dash 30 close parentheses plus open parentheses short dash 24 close parentheses plus 15 Change to subtraction
48 minus 30 minus 24 plus 15 Evaluate
9 Our Solution

summary
The distributive rule is used for the FOIL method when you're multiplying groups of terms in the form open parentheses a plus b close parentheses open parentheses c plus d close parentheses. Remember, these are called binomials. If we're multiplying open parentheses a plus b close parentheses times open parentheses c plus d close parentheses comma we are multiplying two binomials. When we practice using FOIL, the acronym can help us remember the steps for doing that distributing. It is important to remember that FOIL stands for First, Outer, Inner, and Last.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

Terms to Know
FOIL

An acronym to remember the steps for distributing factors in binomial multiplication: first, outside, inside, last.

Formulas to Know
FOIL Method

open parentheses a plus b close parentheses open parentheses c plus d close parentheses equals a c plus a d plus b c plus b d

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