Binomials: What They Are, Parts, Examples, and Operations

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Learn what a binomial is, how to identify its terms, what its parts are, and how to perform basic operations with binomials through clear procedures and step-by-step examples.

Learning Path: Algebra → Algebraic Expressions → Algebraic Terms → Monomials → Binomials

Level: Middle School, High School, Academic Preparation, and Self-Directed Learning

Competency: Identify, interpret, and correctly perform operations with binomials.

What Is a Binomial?

A binomial is an algebraic expression consisting of two terms. The terms are normally separated by addition or subtraction.

x + 5

The expression above contains two terms:

• First term: x
• Second term: 5

Conclusion: x + 5 is a binomial because it contains two terms.

Key idea: The prefix “bi-” means two. Therefore, a binomial contains exactly two terms.

Examples of Binomials

• x + 3 — Two terms
• 4a − 7 — Two terms
• 5m² + 2m — Two terms
• 3x³ − 8y — Two terms
• a²b + 6 — Two terms
• −2p⁴ + 9 — Two terms

How Can You Identify a Binomial?

To identify a binomial, we need to determine how many terms the expression contains.

1. Observe — Look at the complete expression.

2. Separate — Identify the terms.

3. Count — Count the terms.

4. Classify — If there are two terms, it is a binomial.

Example

7x² − 4x

Step 1. Identify the first term: 7x².

Step 2. Identify the second term: −4x.

Result: 7x² − 4x is a binomial.

Parts of a Binomial

Consider the following binomial:

5x² − 3y

It consists of two terms:

Element	Expression	Description
First term	5x²	The first monomial that makes up the binomial.
Second term	−3y	The second monomial, including its sign.
Sign between terms	−	Indicates subtraction between the terms.

Each term in a binomial can, by itself, be a monomial.

Binomial = monomial + monomial, where the second term may have a positive or negative sign.

Binomials with Addition and Binomials with Subtraction

Binomial with Addition

x + 7

Its terms are x and 7.

Binomial with Subtraction

x − 7

Its terms are x and −7.

Important: In an expression such as x − 7, the second term can be identified as −7. The negative sign is part of the term being subtracted.

Like and Unlike Binomials

Binomials are not classified as “like” in exactly the same way as two monomials. What can be determined is whether their corresponding terms are like terms and, more importantly, whether the expressions can be combined when performing operations.

For example:

3x + 5x

Although two terms initially appear, they are like terms and can be combined:

3x + 5x = 8x

The result is no longer a binomial but a monomial.

Combining Like Terms Within Expressions

When an expression contains like terms, we can combine them.

4x + 7x − 2

Step 1. Identify the like terms: 4x and 7x.

Step 2. Add their coefficients: 4 + 7 = 11.

Step 3. Keep x.

4x + 7x − 2 = 11x − 2

Result: After combining like terms, we obtain another binomial.

Adding Binomials

To add binomials, group like terms and simplify the expression.

Example 1

(3x + 5) + (2x + 7)

Step 1. Remove the parentheses while preserving signs:
3x + 5 + 2x + 7

Step 2. Group like terms:
(3x + 2x) + (5 + 7)

Step 3. Add:
5x + 12

(3x + 5) + (2x + 7) = 5x + 12

Example 2

(4a − 3) + (2a + 8)

Step 1: 4a − 3 + 2a + 8

Step 2: (4a + 2a) + (−3 + 8)

Step 3: 6a + 5

(4a − 3) + (2a + 8) = 6a + 5

Subtracting Binomials

When subtracting binomials, it is essential to correctly distribute the negative sign to the second binomial.

(5x + 4) − (2x + 1)

Step 1. Distribute the negative sign:
5x + 4 − 2x − 1

Step 2. Group like terms:
(5x − 2x) + (4 − 1)

Step 3. Simplify:
3x + 3

(5x + 4) − (2x + 1) = 3x + 3

Common mistake: You should not simply remove the second set of parentheses without changing the signs of all its terms when a negative sign appears before it.

Multiplying a Monomial by a Binomial

To multiply a monomial by a binomial, use the distributive property: the monomial is multiplied by each term of the binomial.

Example 1

3x(x + 4)

Step 1. Multiply 3x by x: 3x · x = 3x²

Step 2. Multiply 3x by 4: 3x · 4 = 12x

3x(x + 4) = 3x² + 12x

Example with a Negative Sign

−2a(3a − 5)

Step 1. −2a · 3a = −6a²

Step 2. −2a · (−5) = +10a

−2a(3a − 5) = −6a² + 10a

Multiplying Two Binomials

To multiply two binomials, apply the distributive property to each term.

(x + 2)(x + 3)

Step 1. Multiply x by each term of the second binomial:
x · x + x · 3

Step 2. Multiply 2 by each term:
2 · x + 2 · 3

This gives: x² + 3x + 2x + 6

Step 3. Combine like terms:
3x + 2x = 5x

(x + 2)(x + 3) = x² + 5x + 6

Conjugate Binomials

Two binomials are conjugates when they have the same terms but different signs between them.

(x + 5)   and   (x − 5)

When we multiply them, we obtain a difference of squares:

(x + 5)(x − 5)

Step 1. Apply the distributive property:
x² − 5x + 5x − 25

Step 2. The terms −5x and +5x cancel each other.

(x + 5)(x − 5) = x² − 25

Squaring a Binomial

When a binomial is raised to the second power, it is multiplied by itself.

(x + 3)² = (x + 3)(x + 3)

Step 1. Multiply the binomials:
x · x + x · 3 + 3 · x + 3 · 3 = x² + 3x + 3x + 9

Step 2. Combine like terms:
3x + 3x = 6x

(x + 3)² = x² + 6x + 9

Important: (x + 3)² is not equal to x² + 9. The middle term 6x is necessary.

Common Mistakes with Binomials

Mistake 1: Forgetting to Distribute a Term

In 2x(x + 3), the 2x must be multiplied by both x and 3.

2x² + 6x

Mistake 2: Not Changing All Signs When Subtracting

In (x + 5) − (2x + 3), we must obtain:

x + 5 − 2x − 3

Mistake 3: Confusing the Square of a Binomial with Squares of Its Terms

(x + 4)² ≠ x² + 16

The correct result includes the middle term:

(x + 4)² = x² + 8x + 16

Practice Exercises

Identify which expressions are binomials and, when indicated, perform the operation.

1. 5x + 7

Answer: Binomial.

2. 3a²

Answer: Monomial (not a binomial).

3. 4m − 9

Answer: Binomial.

4. x² + 3x + 2

Answer: Trinomial (not a binomial).

5. Perform addition: (2x + 3) + (4x + 5)

Answer: 2x + 3 + 4x + 5 = 6x + 8

6. Perform subtraction: (7a − 2) − (3a + 4)

Answer: 7a − 2 − 3a − 4 = 4a − 6

7. Perform multiplication: 2x(x + 5)

Answer: 2x² + 10x

8. Multiply binomials: (x + 2)(x + 4)

Answer: x² + 4x + 2x + 8 = x² + 6x + 8

What You Should Remember About Binomials

• Binomial: An algebraic expression consisting of exactly two terms.
• Terms: The individual monomials that make up the binomial.
• Addition and subtraction: Simplify by combining like terms.
• Monomial × binomial: Use the distributive property.
• Binomial × binomial: Multiply each term of the first binomial by each term of the second.
• Binomial squared: Multiply the binomial by itself.

Frequently Asked Questions About Binomials

What Is a Binomial?

It is an algebraic expression consisting of exactly two terms, such as x + 5 or 3x² − 7.

Can a Binomial Have Two Variables?

Yes. For example, 3xy + 5 is a binomial because it contains two terms, even though one of the terms contains two variables.

Can a Binomial Have Like Terms?

Yes. However, if the terms are like terms, they can be combined, and the expression may simplify so that it is no longer a binomial.

What Is a Pair of Conjugate Binomials?

It is a pair of binomials that contains the same terms but opposite signs between them, such as x + 4 and x − 4.

How Do You Multiply a Binomial by a Monomial?

Use the distributive property: multiply the monomial by each term of the binomial.

How Do You Multiply Two Binomials?

Multiply each term of the first binomial by each term of the second binomial, and then combine like terms.

Conclusion

Binomials are algebraic expressions consisting of exactly two terms. Understanding their structure is essential for progressing to operations with polynomials and special products.

To work correctly with binomials, it is necessary to identify their terms, recognize like terms, apply the distributive property, and carefully manage signs.

Mastering these operations provides a foundation for the subsequent study of trinomials, polynomials, special products, and factoring.

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This topic is part of the episode:

S2 E11 | Binomials | Frecuencia Educativa

In this article, we take a deep dive into the topic of Algebraic Terms: what they are, their parts, examples, and like terms.

Listen to the full episode here:

T2 E11 | Monomios, binomios, trinomios y polinomios | Frecuencia Educativa

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Digital Library Classification

Collection: Learning

Knowledge Center: Mathematics

Knowledge Area: Algebra

Topic: Binomials

Title: Binomials: What They Are, Elements, Characteristics, and Examples

Level: Middle School and High School

Content Type: Master Article

Resource Type: Educational Reading

Competencies

• Identify what a binomial is and recognize its elements.
• Recognize the two terms that make up a binomial.
• Distinguish among terms, coefficients, variables, and exponents.
• Determine the degree of a binomial.
• Identify like binomials and distinguish them from expressions with different structures.
• Correctly interpret and represent binomials in algebraic expressions.

Learning Path

Introduction to Algebra → Algebraic Language → Algebraic Expressions → Algebraic Terms → Binomials

Description

Article dedicated to the study of binomials as algebraic expressions consisting of two terms. It explains their definition, elements, characteristics, degree, classification, and examples, as well as the differences between their terms. It provides a foundation for the subsequent study of operations with binomials and special products.