FOIL Method

Learn the FOIL method to multiply two binomials quickly and accurately.

Intermediate25 minLesson
Printable worksheet

Definition

FOIL is a mnemonic that helps you remember the order for multiplying two binomials. Each letter stands for a pair of terms to multiply:
  • First: Multiply the first terms of each binomial
  • Outer: Multiply the outer terms
  • Inner: Multiply the inner terms
  • Last: Multiply the last terms of each binomial
For (a+b)(c+d)(a + b)(c + d):
F: a⋅cO: a⋅dI: b⋅cL: b⋅d\text{F: } a \cdot c \quad \text{O: } a \cdot d \quad \text{I: } b \cdot c \quad \text{L: } b \cdot d
=ac+ad+bc+bd= ac + ad + bc + bd
After multiplying, combine any like terms to simplify your answer.

Try it now

In FOIL, what does the letter "F" stand for?

Worked Examples

Multiply (x+2)(x+3)(x + 2)(x + 3)
1

First: Multiply the first terms

x⋅x=x2x \cdot x = x^2
2

Outer: Multiply the outer terms

x⋅3=3xx \cdot 3 = 3x
3

Inner: Multiply the inner terms

2⋅x=2x2 \cdot x = 2x
4

Last: Multiply the last terms

2⋅3=62 \cdot 3 = 6
5

Combine all terms

x2+3x+2x+6x^2 + 3x + 2x + 6 → x2+5x+6x^2 + 5x + 6

Common Mistakes

Forgetting to multiply ALL four pairs of terms

Why it's wrong: Students sometimes only multiply the first and last terms, missing the middle terms entirely.

Correct: Always follow F-O-I-L in order: First, Outer, Inner, Last. You should have 4 terms before combining.

Sign errors when multiplying negatives

Why it's wrong: Negative times negative equals positive, but students often forget this rule.

Correct: In (x−3)(x−4)(x - 3)(x - 4), the Last step gives (−3)(−4)=+12(-3)(-4) = +12, not −12-12.

Not combining like terms at the end

Why it's wrong: The Outer and Inner terms often produce like terms that should be combined.

Correct: After FOIL, always check for like terms. For (x+2)(x+3)(x+2)(x+3): 3x+2x=5x3x + 2x = 5x.

Confusing x⋅xx \cdot x with x+xx + x

Why it's wrong: The First step is multiplication, not addition.

Correct: x⋅x=x2x \cdot x = x^2, not 2x2x. Remember: multiplying same variables means adding exponents.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

FOIL Method for Binomials (Step-by-Step Example)

by Understand The Math

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Interactive Visual

Area model: multiplying binomials

The rectangle is (x + a) by (x + b). Its four pieces are the four products of FOIL.

x2x^2
3x3x
2x2x
66
xx
33
xx
22

Length × width = area

(x+3)(x+2)=x2+5x+6(x + 3)(x + 2) = x^2 + 5x + 6

Interactive Sandbox

Expression Calculator

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Practice Problems

18 problems
Problem 1 of 18
Easy
In FOIL, what does the letter "F" stand for?

Why It Matters

The FOIL method is essential for algebra and beyond:
  • Factoring: Understanding FOIL helps you factor quadratic expressions in reverse
  • Quadratic Equations: Expanding (x+a)(x+b)(x + a)(x + b) gives you standard form ax2+bx+cax^2 + bx + c
  • Area Problems: Calculating the area of rectangles with variable dimensions
  • Physics and Engineering: Many formulas involve products of binomials
Once you master FOIL, you'll recognize patterns that make factoring much easier!

Real World Applications

Area of a Garden

A rectangular garden has dimensions that can be expressed as binomials.

Example:

If a garden is (x+3)(x + 3) meters long and (x+2)(x + 2) meters wide, its area is (x+3)(x+2)=x2+5x+6(x+3)(x+2) = x^2 + 5x + 6 square meters.

1Try It Yourself

A rectangular pool has length (x+5)(x + 5) meters and width (x+1)(x + 1) meters.

What is the area of the pool in expanded form?

Step 1: Write the mathematical expression

Use FOIL to multiply (x+5)(x+1)(x + 5)(x + 1):

Projectile Motion

In physics, the path of a thrown object often involves products of binomials.

Example:

If height depends on (t+2)(t−5)(t + 2)(t - 5) where tt is time, expanding gives t2−3t−10t^2 - 3t - 10 which is easier to analyze.

2Try It Yourself

A ball's trajectory involves the expression (t+4)(t−1)(t + 4)(t - 1) where tt is time in seconds.

Expand this expression to standard form.

Step 1: Write the mathematical expression

Apply FOIL to (t+4)(t−1)(t + 4)(t - 1):

Key Takeaways

  • 1FOIL stands for First, Outer, Inner, Last - the order to multiply binomial terms
  • 2First: multiply the first terms of each binomial
  • 3Outer: multiply the outer terms, Inner: multiply the inner terms
  • 4Last: multiply the last terms of each binomial
  • 5After multiplying all four pairs, combine any like terms (usually the O and I terms)

Frequently Asked Questions

FOIL only works for multiplying two binomials (expressions with exactly 2 terms each). For other polynomials, use the distributive property to multiply each term in one polynomial by every term in the other.
FOIL only works for multiplying two binomials (expressions with exactly 2 terms each). For other polynomials, use the distributive property to multiply each term in one polynomial by every term in the other.
In (x+a)(x+b)(x + a)(x + b), the Outer gives bxbx and the Inner gives axax. Both are xx terms (like terms), so they combine to (a+b)x(a + b)x.
Yes! (x+a)(x+b)=x2+(a+b)x+ab(x + a)(x + b) = x^2 + (a+b)x + ab. The middle coefficient is the sum of aa and bb, and the constant is their product.

Glossary

Binomial
An algebraic expression with exactly two terms, such as x+3x + 3 or 2x−52x - 5
FOIL
A mnemonic for multiplying two binomials: First, Outer, Inner, Last
Like terms
Terms with the same variable raised to the same power, such as 3x3x and 5x5x
Trinomial
An algebraic expression with exactly three terms, often the result of FOILing two binomials

Formula Card

FOIL Pattern

(a+b)(c+d)=ac+ad+bc+bd(a + b)(c + d) = ac + ad + bc + bd

Multiply First, Outer, Inner, Last terms, then combine like terms

Special Case

(x+a)(x+b)=x2+(a+b)x+ab(x + a)(x + b) = x^2 + (a+b)x + ab

For binomials with the same first term, the pattern simplifies

Today's challengeOne problem, three tries

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