An example of FOIL method with answer shows how to multiply two binomials by organizing four products: First, Outer, Inner, and Last. The FOIL method is a useful shortcut for expressions such as (x + 4)(x + 2), but it works because of the more general distributive property. Once the four products are found, combining like terms produces the simplified answer.
What Is the FOIL Method?
FOIL is a mnemonic used to multiply two binomials, which are algebraic expressions containing two terms. Each binomial usually has the form ax + b, where a and b are constants and x is a variable.
The four labels describe where each multiplication comes from:
- F — First: Multiply the first term in each binomial.
- O — Outer: Multiply the first term of the first binomial by the last term of the second binomial.
- I — Inner: Multiply the last term of the first binomial by the first term of the second binomial.
- L — Last: Multiply the last term in each binomial.
For two binomials, the general pattern is:
(a + b)(c + d) = ac + ad + bc + bd
This pattern is not a new mathematical rule. It is simply the distributive property applied twice:
(a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd
FOIL helps students remember every required multiplication and avoid missing a term.
How to Use the FOIL Method
To multiply binomials with FOIL, follow these steps:
- Identify the two binomials. Make sure each expression has exactly two terms.
- Multiply the First terms.
- Multiply the Outer terms.
- Multiply the Inner terms.
- Multiply the Last terms.
- Write all four products.
- Combine like terms when possible.
- Check the answer by expanding the result in the opposite direction.
The method is especially useful for quadratic expressions. A product of two linear binomials commonly produces a quadratic expression with a squared term, a linear term, and a constant term Small thing, real impact. That alone is useful..
Example 1: Simple Positive Binomials
Problem
Expand:
(x + 3)(x + 5)
Step-by-Step Answer
Apply each part of FOIL:
- First: x · x = x²
- Outer: x · 5 = 5x
- Inner: 3 · x = 3x
- Last: 3 · 5 = 15
Now add the four products:
x² + 5x + 3x + 15
The terms 5x and 3x are like terms, so combine them:
x² + 8x + 15
Therefore:
(x + 3)(x + 5) = x² + 8x + 15
Check
Distribute each term in the first binomial:
x(x + 5) + 3(x + 5)
x² + 5x + 3x + 15
x² + 8x + 15
The result matches the FOIL answer.
Example 2: A Binomial with a Negative Term
Problem
Expand:
(x − 4)(x + 7)
Step-by-Step Answer
Multiply the terms according to FOIL:
- First: x · x = x²
- Outer: x · 7 = 7x
- Inner: *−4 · x
= −4x*
- Last: −4 · 7 = −28
Combine the products:
x² + 7x − 4x − 28
The like terms are 7x and −4x:
x² + 3x − 28
Therefore:
(x − 4)(x + 7) = x² + 3x − 28
Check
Distribute each term in the first binomial:
x(x + 7) − 4(x + 7)
x² + 7x − 4x − 28
x² + 3x − 28
The answer is correct.
Example 3: Binomials with Coefficients
Problem
Expand:
(2x + 3)(4x − 5)
Step-by-Step Answer
Apply FOIL:
- First: 2x · 4x = 8x²
- Outer: 2x · −5 = −10x
- Inner: 3 · 4x = 12x
- Last: 3 · −5 = −15
Add the four products:
8x² − 10x + 12x − 15
Combine the like terms −10x and 12x:
8x² + 2x − 15
Therefore:
(2x + 3)(4x − 5) = 8x² + 2x − 15
Example 4: Two Negative Terms
Problem
Expand:
(3x − 2)(5x − 4)
Step-by-Step Answer
Use FOIL:
- First: 3x · 5x = 15x²
- Outer: 3x · −4 = −12x
- Inner: −2 · 5x = −10x
- Last: −2 · −4 = 8
Now combine:
15x² − 12x − 10x + 8
The like terms are −12x and −10x:
15x² − 22x + 8
Therefore:
(3x − 2)(5x − 4) = 15x² − 22x + 8
Example 5: Difference of Squares
Problem
Expand:
(x + 6)(x − 6)
Step-by-Step Answer
Apply FOIL:
- First: x · x = x²
- Outer: x · −6 = −6x
- Inner: 6 · x = 6x
- Last: 6 · −6 = −36
Add the products:
x² − 6x + 6x − 36
The middle terms cancel because −6x + 6x = 0:
x² − 36
Therefore:
(x + 6)(x − 6) = x² − 36
This is an example of the difference of squares pattern:
(a + b)(a − b) = a² − b²
Example 6: Perfect Square Trinomial
Problem
Expand:
(x + 4)²
Step-by-Step Answer
First rewrite the square as a product of two identical binomials:
(x + 4)(x + 4)
Now apply
