Binomials are the building blocks of polynomial expressions in algebra. They are simple expressions containing exactly two terms combined by addition (+) or subtraction (-). Multiplying binomials involves using methods such as the distributive property, FOIL method, or vertical method.

In this article, You will get to know about Binomials what it is, How to multiply Binomial, and examples of methods that are used for multiplication like the Distributive property, the FOIL method and the Vertical method, there are also available practice questions and examples to understand better.

What is a Binomial?

In algebra, a binomial is a polynomial that consists of exactly two terms. These terms can be monomials (a single term consisting of a coefficient and a variable raised to a power or just a number) and they are combined using addition or subtraction.

Here’s a breakdown of key points about binomials:

• Number of Terms: A defining characteristic of a binomial is that it has only two terms.
• Types of Terms: The terms in a binomial can be variables raised to any power (including exponents of zero or one), coefficients (numerical factors), or a combination of both.
• Operations: Binomials are formed by adding or subtracting these two terms.
• Monomials: Single terms with a numerical coefficient multiplied by a variable raised to a non-negative whole number exponent (for example = 3x^2, -5y).
• Constants: Numerical values alone (for example = 7, -2).

Here are some examples of binomials:

• 3x + 5 (where 3x is a monomial and 5 is a constant term)
• x^2 – y (where x^2 and y are monomials with different variables and exponents)
• -2a + b (where -2a is a monomial with a negative coefficient)

How to multiply Binomials?

Multiplication of Binomials is similar to the simple multiplication of any two digits but the difference is that binomial multiplication uses the concept of multiplication of algebraic expressions. The first step is that one binomial term is multiplied by the other binomial term after that the algebraic sum of that product is taken and then different methods of binomial multiplication are applied like the distributive property, the FOIL method and the Vertical method.

Multiplying binomials is a fundamental operation in algebra. There are three main methods are as following :

1. Distributive Property
2. FOIL Method
3. Vertical Method

The Distributive Property

The distributive property is a fundamental mathematical rule that allows us to distribute a factor across a sum (or difference). In simpler terms, it tells us that multiplying a single number by a sum of terms is the same as multiplying that number by each term in the sum individually and then adding the products together.

Multiplying Binomials using the Distributive Property

Binomials are expressions with two terms. Let’s see how we can multiply two binomials using the distributive property:

Example:

Multiply the following binomial: (a + b) × (x + y)

Step-by-Step Solution:

Step 1.) Identify the binomials: We have two binomials, (a + b) and (x + y).

Step 2.) Distribute the first term of the first binomial: Apply the distributive property with “a” (the first term of the first binomial) being distributed across “(x + y)”.

a × (x + y) = (a × x) + (a × y) // Distribute “a”

Step 3.) Distribute the second term of the first binomial: Do the same for “b” (the second term of the first binomial).

b × (x + y) = (b × x) + (b × y) // Distribute “b”

Step 4.) Combine like terms: Now we have four terms: (a x x), (a x y), (b x x), and (b x y). Since we’re multiplying variables, their order doesn’t affect the answer (commutative property). So we can group the terms based on the variables:

Combine terms with “x”: (a x x) + (b x x) = (a + b) x x

Combine terms with “y”: (a x y) + (b x y) = (a + b) x y

Step 4.) Put it all together: Finally, we combine the terms we got in step 4 using the distributive property again: ((a + b) x x) + ((a + b) x y) = (a + b) (x + y)

Solution: The product of the two binomials (a + b) and (x + y) is (a + b) (x + y) by using the distributive property.

The FOIL Method

The FOIL method provides a systematic approach to multiplying binomials. It’s an acronym that stands for First, Outer, Inner, Last. The FOIL method expands on the distributive property. By multiplying each term from one binomial with each term from the other binomial.

FOIL Method Breakdown:

1. First: Multiply the first terms of each binomial.
2. Outer: Multiply the outer terms of each binomial (the terms that aren’t next to each other in the parentheses).
3. Inner: Multiply the inner terms of each binomial (the terms that are next to each other in the parentheses).
4. Last: Multiply the last terms of each binomial.

Example: Multiplying Binomials using FOIL

Let’s multiply the binomials (x + 2) and (x + 3) using the FOIL method:

Steps:

1. First: (x) * (x) = x²
2. Outer: (x) * (3) = 3x
3. Inner: (2) * (x) = 2x
4. Last: (2) * (3) = 6

Combining Like Terms:

Now, we have four terms: x², 3x, 2x, and 6. Since we’re multiplying variables, the order doesn’t affect the answer (commutative property). So we can simply combine the x terms:

x² + (3x + 2x) + 6 = x² + 5x + 6

Solution: The product of (x + 2) and (x + 3) is x² + 5x + 6 using the FOIL method.

The Vertical Method

The Vertical method is another way to multiply binomials. Multiplying Binomials using the vertical method is similar to the vertical multiplication of whole numbers this method applies to all polynomial multiplications.

Vertical Method for Multiplying Binomials:

• Set up the binomials: Write the two binomials one on top of the other aligning the like terms (terms with the same variable) in separate columns.

• Multiply and distribute: Multiply each term from the top binomial with each term from the bottom binomial. Write the product of each multiplication term diagonally below the line aligning it with the corresponding variable in the answer.

• Add the products: Draw a horizontal line to separate the products from the original terms. Then, add the corresponding terms in each column vertically just like you would add regular numbers.

Example: Multiplying Binomials Vertically

Let’s multiply the same binomials from previous examples, (x + 2) and (x + 3), using the vertical method:

Steps:

Step 1. )Set up the binomial:

x + 2

x + 3

Step 2. ) Multiply and distribute:

x + 2

x + 3

———–

x^2 (x * x)

3x (x * 3)

2x (2 * x)

6 (2 * 3)

Step 3.) Add the products:

x + 2

x + 3

——-

x^2 (x * x)

• 3x (x * 3)
• 2x (2 * x)

————————

x^2 + 5x + 6

Solution : The product of (x + 2) and (x + 3) is x² + 5x + 6 using the Vertical method.

Multiplying Binomial Examples: Solved

Example 1: Distributive Property (Simple Binomials)

Multiply: (x + 2) (x + 3)

Solution:

1. Distribute (x + 2) across (x + 3): (x + 2) * (x + 3) = (x * (x + 3)) + (2 * (x + 3))
2. Expand the products: = (x^2 + 3x) + (2x + 6)
3. Combine like terms: = x^2 + (3x + 2x) + 6 = x^2 + 5x + 6

Example 2: FOIL Method (Binomials with Variables)

Multiply: (2a – 3) (b + 1)

Solution:

Follow the FOIL method:

• First: (2a) * (b) = 2ab
• Outer: (2a) * (1) = 2a
• Inner: (-3) * (b) = -3b
• Last: (-3) * (1) = -3

Combine terms: 2ab + 2a – 3b – 3

Example 3: Binomial with Different Variable Names

Multiply: (2a + b) (c – d)

Solution (FOIL method is efficient here):

Follow the FOIL method:

• First: (2a) * (c) = 2ac
• Outer: (2a) * (-d) = -2ad
• Inner: (b) * (c) = bc
• Last: (b) * (-d) = -bd

Combine terms: 2ac – 2ad + bc – bd

Example 4: Negative Coefficients

Multiply: (a – 2) (3a + 5)

Solution (Distributive property works well here):

1. Distribute (a – 2) across (3a + 5): (a – 2) * (3a + 5) = (a * (3a + 5)) + (-2 * (3a + 5))
2. Expand the products: = (3a^2 + 5a) + (-6a – 10)
3. Combine like terms: 3a^2 – a – 10

Example 5: Binomial with a Squared Term and a Constant (Vertical Method)

Multiply: (y^2 – 1) (y + 2)

Solution :

Set up the binomials:

y^2 – 1

y + 2

——-

Multiply and distribute:

y^2 – 1

y + 2

————–

y^3 (y^2 * y) = -2y (-1 * 2y) -y^2 (y^2 * -1) +2 (-1 * 2)

Add the products:

y^2 – 1 y + 2

y^3 – 3y + 2 (combine y^2 terms)

Multiplying Binomial Practice Questions: Unsolved

Below is the list of 10 practice questions for multiplying Binomials are as follows :