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. For example, 3(5 + 9) can be evaluated using distribution. The 3 outside of (5 + 9) is multiplied into each term, written equivalently as (3•5)+(3•9). We can evaluate this as the sum of 15 and 27, or 42. We can confirm this by evaluating 3(5 + 9) using the order of operations: 3(5 + 9) = 3(14) = 42.
The FOIL Method
Earlier, we saw how distribution can help us evaluate expressions in the form a(b + c), by distributing a into b + c to get ab + ac. How could we distribute something like (4 + 3)(5 + 1)?
We can evaluate such expressions by distributing, 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:
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). Our next step was to multiply the two outer terms. 4 and 1 are the outermost terms in our expression. In the next step, we multiplied the two inner terms: 3 and 5. And 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).
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.
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!
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!
An acronym to remember the steps for distributing factors in binomial multiplication: first, outside, inside, last.