Algebraic Terms: Parts, Examples, Like Terms, and Degree

Español | English

An algebraic term is one of the basic elements used to build algebraic expressions. Understanding its parts makes it easier to read, interpret, simplify, and operate with algebraic expressions.

In this article, we will identify the sign, coefficient, variable part, and exponents of an algebraic term. We will also learn how to determine its degree, recognize like terms, combine them correctly, and avoid common mistakes.

Learning sequence:

Introduction to Algebra → Algebraic Language → Algebraic Expressions → Algebraic Terms → Monomials and Polynomials

What Is an Algebraic Term?

An algebraic term is an expression that consists of numerical factors and variables with their exponents, connected by multiplication.

5x −3x² 7xy² −4m²n³

Each of these is a single algebraic term.

Important distinction: In the school context, a monomial is an algebraic expression consisting of a single term, such as 5x, −3x², or 7xy².

Algebraic Term vs. Algebraic Expression

A single term can be part of a larger algebraic expression. Terms in an expression are separated at the main level by addition or subtraction.

5x² − 3x + 8

This expression contains three terms:

5x²
−3x
8

Notice that multiplication does not separate terms. For example, 6xy is one term because the factors are multiplied.

Parts of an Algebraic Term

An algebraic term can be analyzed by identifying four important elements: the sign, coefficient, variable part, and exponents.

1. Sign

The sign indicates whether the term is positive or negative.

+5x² −5x²

If no negative sign appears, the term is normally understood to be positive.

2. Coefficient

The coefficient is the numerical factor multiplying the variable part.

7a²b³

The coefficient is 7, and the variable part is a²b³.

When a variable has no visible numerical coefficient, its coefficient is understood to be 1.

x = 1x

Similarly:

−x = −1x

Depending on the mathematical convention being used, the negative sign may be described separately from the numerical coefficient, or the signed coefficient may be described as a whole. Both descriptions represent the same mathematical term.

3. Variable Part

The variable part contains the letters and their exponents.

−5x²y³

The variable part is:

x²y³

The letters x and y represent variables.

4. Exponents

An exponent indicates how many times a variable is used as a factor.

x³ = x · x · x

Therefore, in the term −12m³n², the exponent of m is 3 and the exponent of n is 2.

How to Identify the Parts of a Term

Let us analyze the following term:

−5x²y³

Step 1. Identify the sign: the sign is negative (−).

Step 2. Identify the coefficient: the numerical coefficient is 5; if the sign is included, the signed coefficient is −5.

Step 3. Identify the variable part: x²y³.

Step 4. Identify the exponents: x has exponent 2 and y has exponent 3.

Degree of an Algebraic Term

For a monomial with nonnegative integer exponents, the degree of the term is found by adding the exponents of all its variables.

Example 1

6x⁴

There is one variable with exponent 4.

Degree = 4

Example 2

3x²y³

Degree = 2 + 3 = 5

Example 3

−7a³b²c

The variable c has an implicit exponent of 1.

Degree = 3 + 2 + 1 = 6

Common mistake: Do not use the coefficient when calculating the degree. The degree depends on the exponents of the variables, not on the numerical coefficient.

Like Terms

Two or more terms are like terms when they have exactly the same variable part, including the same variables with the same exponents.

Examples of Like Terms

3x² and 8x²

4x²y³ and −9x²y³

6a²b and −2ba²

The last pair is also like terms because multiplication is commutative:

a²b = ba²

Examples of Unlike Terms

5x² and −3x

They have different exponents of x.

4xy and 7xz

They contain different variables.

How to Combine Like Terms

When combining like terms, add or subtract their coefficients while keeping the common variable part unchanged.

Example 1

3x + 5x = (3 + 5)x = 8x

Example 2

9x² − 4x² = (9 − 4)x² = 5x²

Example 3

7ab² + 2ab² = (7 + 2)ab² = 9ab²

Important: Only like terms can be combined by adding or subtracting their coefficients. For example, 3x + 5y cannot be simplified to 8xy or 8x.

Common Mistakes

Confusing the coefficient with the exponent.
Forgetting that a variable without a visible coefficient has coefficient 1.
Forgetting that a variable without a visible exponent has exponent 1.
Combining terms that are not like terms.
Ignoring the negative sign of a term.
Adding exponents when combining like terms. The exponents remain unchanged when like terms are added or subtracted.

Practice Exercises

1. Identify the coefficient, variable part, and exponent in −8x³.

Answer: coefficient −8; variable part x³; exponent 3.

2. Identify the coefficient and variable part in 5a²b⁴.

Answer: coefficient 5; variable part a²b⁴.

3. Determine the degree of 7x³y².

Answer: 3 + 2 = 5.

4. Are 4x² and −9x² like terms?

Answer: Yes. They have the same variable part, x².

5. Are 4x² and −9x like terms?

Answer: No. Their exponents are different.

6. Simplify: 6x + 2x.

Answer: 8x.

7. Simplify: 10a² − 3a².

Answer: 7a².

8. Identify the like terms in 3x² + 5x − 7x² + 4.

Answer: 3x² and −7x².

Frequently Asked Questions

What is an algebraic term?

An algebraic term is a mathematical expression consisting of numerical factors and variables with their exponents, connected by multiplication.

What is the coefficient of x?

The coefficient of x is 1 because x can be written as 1x.

What is the coefficient of −x?

The signed coefficient is −1 because −x can be written as −1x.

How do I find the degree of a term?

Add the exponents of all the variables in the term. For example, the degree of 4x²y³ is 2 + 3 = 5.

When are two terms like terms?

They are like terms when they have exactly the same variables with the same exponents. Their coefficients may be different.

Can unlike terms be combined?

No. Unlike terms cannot be combined by simply adding or subtracting their coefficients.

What Comes Before and After Algebraic Terms?

Algebraic Terms is part of a progressive learning sequence in the study of algebra.

Before: Algebraic Expressions — students learn what algebraic expressions are, how they are formed, and how their terms are identified.

Current topic: Algebraic Terms — students examine the structure of an individual term, including its sign, coefficient, variable part, exponents, degree, and relationship to like terms.

After: Monomials and Polynomials — students move from individual terms to expressions classified according to the number and structure of their terms.

Introduction to Algebra → Algebraic Language → Algebraic Expressions → Algebraic Terms → Monomials and Polynomials

Conclusion

Understanding algebraic terms is fundamental for progressing through algebra. Identifying the sign, coefficient, variable part, and exponents allows students to interpret algebraic notation accurately and prepares them for operations with monomials and polynomials.

The ability to recognize like terms is especially important because it establishes the foundation for simplifying algebraic expressions. The key is to compare the variable parts carefully: the variables and their exponents must match before coefficients can be combined.

TECHNICAL RESOURCE SHEET

Collection: Mathematics

Knowledge Center: Algebra

Knowledge Area: Algebraic Foundations and Language

Topic: Algebraic Terms

Educational Level: Middle School / High School

Resource Type: Master Article

Description:
An educational resource dedicated to the study of algebraic terms and the elements that make them up. It progressively explains the sign, coefficient, variable part, exponents, and degree of a term, while also introducing the identification and simplification of like terms through examples and exercises.

Competencies:

Identify the elements of an algebraic term.
Distinguish between the sign, coefficient, variable part, and exponents.
Determine the degree of an algebraic term.
Recognize like and unlike terms.
Correctly combine like terms.
Interpret the structure of algebraic expressions.
Apply algebraic notation accurately.

Learning Path:
Algebra 1. Foundations and Algebraic Language

Previous Article:
Algebraic Expressions

Current Article:
Algebraic Terms

Next Article:
Monomials and Polynomials

Classification Hierarchy:
Mathematics → Algebra → Algebraic Foundations and Language → Algebraic Terms