FOIL
Method
A technique for distributing two binomials.
The letters FOIL stand for First, Outer, Inner, Last. First means
multiply the terms which
occur first in each binomial. Then Outer means multiply the outermost
terms in the product. Inner means multiply the innermost
two terms.
Last means multiply the terms which occur last in each binomial.
Then simplify the products and combine any like
terms which may
occur.
Example:
(x +
2)(x +
5) = x·x + x·5 +
2·x + 2·5
First Outer Inner Last
= x2 +
7x + 10
See
also
Distributing
rules, binomial theorem
Worked Example
Problem: Expand and simplify (3x + 4)(2x − 5).
First: Multiply the first terms of each binomial: 3x and 2x.
3x⋅2x=6x2 Outer: Multiply the outermost terms: 3x and −5.
3x⋅(−5)=−15x Inner: Multiply the innermost terms: 4 and 2x.
4⋅2x=8x Last: Multiply the last terms of each binomial: 4 and −5.
4⋅(−5)=−20 Combine: Add all four products together and combine the like terms −15x and 8x.
6x2−15x+8x−20=6x2−7x−20 Answer: (3x+4)(2x−5)=6x2−7x−20 Another Example
Problem: Expand and simplify (x − 3)(x − 7).
First: Multiply the first terms: x and x.
x⋅x=x2 Outer: Multiply the outer terms: x and −7.
x⋅(−7)=−7x Inner: Multiply the inner terms: −3 and x.
(−3)⋅x=−3x Last: Multiply the last terms: −3 and −7. Remember that a negative times a negative gives a positive.
(−3)(−7)=21 Combine: Add the four products and combine like terms −7x and −3x.
x2−7x−3x+21=x2−10x+21 Answer: (x−3)(x−7)=x2−10x+21 Frequently Asked Questions
Does FOIL work for multiplying a binomial by a trinomial?
No. FOIL is specifically designed for multiplying two binomials (two terms × two terms). When one or both factors have more than two terms, you need the general distributive property: multiply every term in the first factor by every term in the second factor. FOIL is really just a special case of this general rule.
Does the order of FOIL matter?
The order in which you compute the four products does not change the final answer, because addition is commutative. FOIL simply provides a systematic sequence so you don't accidentally skip a product. As long as you compute all four products and combine like terms correctly, any order works.
FOIL Method vs. Distributive Property
FOIL is a specific application of the distributive property that applies only to the product of two binomials (four multiplications). The distributive property is the general rule that works for any number of terms in each factor. For example, multiplying a binomial by a trinomial requires six multiplications — FOIL cannot handle this, but the distributive property can. Think of FOIL as a convenient shortcut for one particular case of distribution.
Why It Matters
Multiplying two binomials is one of the most common operations in algebra, appearing in factoring, solving quadratic equations, and simplifying rational expressions. The FOIL method gives you a reliable, step-by-step routine that prevents you from missing any of the four required products. Mastering it also builds your intuition for the general distributive property, which you will use throughout higher-level mathematics.
Common Mistakes
Mistake: Forgetting the Inner or Outer product and writing only two terms instead of four.
Correction: Always compute all four products — First, Outer, Inner, Last — before combining like terms. Two binomials produce four individual products every time.
Mistake: Losing a negative sign during multiplication, especially in the Last step when both terms are negative.
Correction: Track signs carefully at each step. A negative times a negative gives a positive, and a negative times a positive gives a negative. Write out each product with its sign before simplifying.
