Multiplying Binomials

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Multiplying binomials means multiplying each term of the first binomial by each term of the second, then combining like terms. The FOIL method helps you remember all four products: First, Outer, Inner, Last. The two middle products are often like terms, so most answers end up with three terms. For example, (x + 3)(x + 5) = x^2 + 8x + 15.

Before this lesson: You should know the distributive property, exponent rules, and adding and subtracting polynomials.

What Does It Mean to Multiply Binomials?

A binomial is a polynomial with two terms, like x + 3 or 2x - 5. To multiply two binomials, every term in the first one must multiply every term in the second one. That makes 4 products.

An area model shows why. Picture a rectangle that is x + 3 tall and x + 5 wide. Split it into 4 smaller rectangles. The whole area is the sum of the 4 pieces.

A rectangle split into four parts. The top side is labeled x and 5, and the left side is labeled x and 3. The four parts are x², 5x, 3x and 15. Beside it: (x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15.
The four pieces have areas x², 5x, 3x and 15. Add them and combine like terms to get x² + 8x + 15.

The two middle pieces, 5x and 3x, are like terms. That is why most answers have 3 terms, not 4.

How to Multiply Binomials

Every method uses the same rule. It is the distributive property used twice:

(a + b)(c + d) = ac + ad + bc + bd
MethodHow it worksBest for
FOILMultiply the First terms, Outer terms, Inner terms and Last terms.Two binomials, done fast
DistributiveDistribute the first term, then the second term, across the other binomial.Seeing why it works
Box methodPut one binomial across the top and one down the side. Multiply to fill each box.Keeping signs organized

Steps (FOIL)

  1. First: multiply the first term of each binomial.
  2. Outer: multiply the two terms on the outside.
  3. Inner: multiply the two terms on the inside.
  4. Last: multiply the last term of each binomial.
  5. Combine like terms. Usually the outer and inner products combine.

Keep the sign with each term. In (x - 4), the second term is -4, not 4.

The product (2x + 3)(x − 4) with four arcs joining terms. Above: First joins 2x and x, Outer joins 2x and −4. Below: Inner joins 3 and x, Last joins 3 and −4. The products are 2x², −8x, 3x and −12, which combine to 2x² − 5x − 12.
FOIL makes sure you get all four products. Then combine the outer and inner products.

Worked Examples

Example 1: Two simple binomials

Multiply (x + 3)(x + 5).

\begin{aligned} (x + 3)(x + 5) &= \underbrace{x \cdot x}_{\text{F}} + \underbrace{x \cdot 5}_{\text{O}} + \underbrace{3 \cdot x}_{\text{I}} + \underbrace{3 \cdot 5}_{\text{L}} \\ &= x^2 + 5x + 3x + 15 \\ &= x^2 + 8x + 15 \end{aligned}

Check with x = 2: (5)(7) = 35 and 4 + 16 + 15 = 35.

Example 2: Negative numbers

Multiply (x - 6)(x + 2).

\begin{aligned} (x - 6)(x + 2) &= x^2 + 2x - 6x - 12 \\ &= x^2 - 4x - 12 \end{aligned}

The last product is (-6)(2) = -12. A negative times a positive is negative.

Example 3: A coefficient on x

Multiply (2x + 3)(x - 4).

FOIL
2x \cdot x = 2x^22x \cdot (-4) = -8x3 \cdot x = 3x3 \cdot (-4) = -12

Combine the middle terms: -8x + 3x = -5x. The answer is 2x^2 - 5x - 12.

Example 4: The box method with two variables

Multiply (3a - 2b)(a - 5b).

×a-5b
3a3a^2-15ab
-2b-2ab10b^2

Add the four boxes: 3a^2 - 15ab - 2ab + 10b^2 = 3a^2 - 17ab + 10b^2. The box for (-2b)(-5b) is positive, because a negative times a negative is positive.

Example 5: Word problem

A square garden has sides of x feet. The owner makes it 4 feet longer and 1 foot narrower. Write a polynomial for the new area. Then find the new area when x = 10.

\begin{aligned} A &= (x + 4)(x - 1) \\ &= x^2 - x + 4x - 4 \\ &= x^2 + 3x - 4 \end{aligned}

When x = 10: A = 100 + 30 - 4 = 126 square feet. Check: the garden is 14 feet by 9 feet, and 14 \cdot 9 = 126.

Common Mistakes

MistakeExample of the errorFix
Squaring each term(x + 3)^2 = x^2 + 9Write it as (x + 3)(x + 3) and use FOIL: x^2 + 6x + 9.
Skipping the outer or inner product(x + 2)(x + 7) = x^2 + 14There are always 4 products: x^2 + 7x + 2x + 14 = x^2 + 9x + 14.
Losing a negative sign(x - 6)(x + 2) = x^2 - 4x + 12(-6)(2) = -12, so the answer is x^2 - 4x - 12.
Forgetting the exponent2x \cdot x = 2xx \cdot x = x^2, so 2x \cdot x = 2x^2.
Not combining like termsx^2 + 5x + 3x + 15Combine the middle terms: x^2 + 8x + 15.

Choosing a Method and What Comes Next

All three methods give the same answer, so use the one you trust most. FOIL is fast for two binomials. The box method is great when there are negatives, because each sign sits in its own box.

FOIL only works for two binomials. To multiply bigger polynomials, like a binomial times a trinomial, you use the distributive property or a bigger box. That is the next lesson, Multiplying Polynomials.

Some products come up so often that they have shortcuts, like (x + 3)^2 and (x + 3)(x - 3). You will learn these special products later in this unit. Later, you will also run FOIL backward to factor.

Practice Problems

  1. Multiply (x + 2)(x + 7).
  2. Multiply (x - 3)(x - 8).
  3. Multiply (x + 5)(x - 9).
  4. Multiply (y - 4)(y + 4).
  5. Multiply (3x + 1)(x + 6).
  6. Multiply (2x - 5)(3x + 2).
  7. Multiply (4m - 3)(2m - 1).
  8. Multiply (x + 6)^2.
  9. Multiply (2a + b)(a - 3b).
  10. A rug is 2x + 1 feet long and x + 3 feet wide. Write its area as a polynomial. Then find the area when x = 4.
Show answers
  1. x^2 + 7x + 2x + 14 = x^2 + 9x + 14
  2. x^2 - 8x - 3x + 24 = x^2 - 11x + 24
  3. x^2 - 9x + 5x - 45 = x^2 - 4x - 45
  4. y^2 + 4y - 4y - 16 = y^2 - 16
  5. 3x^2 + 18x + x + 6 = 3x^2 + 19x + 6
  6. 6x^2 + 4x - 15x - 10 = 6x^2 - 11x - 10
  7. 8m^2 - 4m - 6m + 3 = 8m^2 - 10m + 3
  8. (x + 6)(x + 6) = x^2 + 6x + 6x + 36 = x^2 + 12x + 36
  9. 2a^2 - 6ab + ab - 3b^2 = 2a^2 - 5ab - 3b^2
  10. (2x + 1)(x + 3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3. At x = 4: 32 + 28 + 3 = 63 square feet. Check: the rug is 9 feet by 7 feet, and 9 \cdot 7 = 63.

FAQ

What does FOIL stand for?

FOIL stands for First, Outer, Inner, Last. It is a reminder to multiply all four pairs of terms when you multiply two binomials.

Do you have to use FOIL?

No. FOIL is just the distributive property in a set order. The box method or distributing twice gives the same answer.

Why isn’t (x + 3)² equal to x² + 9?

Squaring means multiplying (x + 3)(x + 3). FOIL gives x^2 + 3x + 3x + 9 = x^2 + 6x + 9. The middle term 6x comes from the outer and inner products.

What is the box method?

Write one binomial across the top of a 2-by-2 grid and the other down the side. Multiply to fill each box, then add the boxes and combine like terms.

Summary

  • To multiply binomials, multiply each term in the first by each term in the second. That gives 4 products.
  • FOIL means First, Outer, Inner, Last.
  • The box method and distributing twice give the same answer.
  • Keep each sign with its term, and remember x \cdot x = x^2.
  • Combine like terms at the end. The outer and inner products usually combine.

Previous lesson: Adding and Subtracting Polynomials

Next lesson: Multiplying Polynomials

Back to the Algebra 1 Index

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