The FOIL method lets you multiply two binomials in a particular order. You don't have to multiply binomials by following the FOIL order, but it does make the process easier. The letters in FOIL refer to two terms (one from each of two binomials) multiplied together in a certain order: First, Outer, Inner, and Last.
Example 1: (2x + 3)(3x – 1)
The following steps demonstrate how to use FOIL on this multiplication problem.
Multiply the first term of each binomial together.
Multiply the outer terms together.
(2x)(–1) = –2x
Multiply the inner terms together.
(3)(3x) = 9x
Multiply the last term of each expression together.
(3)(–1) = –3
List the four results of FOIL in order.
Combine the like terms.
Example 2: (x – 3)(2x – 9)
See how the FOIL numbered steps work on a couple of negative terms.
Multiply the first terms.
Multiply the outer terms.
(x)(–9) = –9x
Multiply the inner terms.
(–3)(2x) = –6x
Multiply the last terms.
(–3)(–9) = 27
List the four results of FOIL in order.
Combine the like terms.
Example 3: [x + (y – 4)][3x + (2y + 1)]
This example is a bit more complicated, but FOIL makes it much easier. The tasks are broken down into smaller, simpler steps, and then the results are combined.
Multiply the first terms.
Multiply the outer terms.
(x)(2y + 1) = 2xy + x
Multiply the inner terms.
(y – 4)(3x) = 3xy – 12x
Multiply the last terms.
The last terms are also two binomials. You FOIL these binomials when you finish this series of FOIL steps.
(y – 4)(2y + 1)
List the four results of FOIL in order.
Combine like terms.
FOIL the product of two binomials from Step 4: (y – 4)(2y + 1).
Multiply the outer terms: (y)(1) = y
Multiply the inner terms: (–4)(2y) = –8y
Multiply the last terms: (–4)(1) = –4
Replace the two binomials multiplied together with this new result, and then rewrite the entire problem.
