Monomial
2026-02-28 12:40 Diff
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Last updated on October 21, 2025

In algebra, a monomial is an expression with only one term. It may consist of one or more variables, a constant, and the product of both. In this article, we will discuss monomials, their components, methods to identify them, and how to factorize them with examples.

What are Monomials?

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The word “mono” means one, so expressions with a single term are called monomials. A monomial consists of a coefficient, one or more variables, and non-negative integer exponents. The degree of a monomial is the sum of the exponents of its variables. For example, in \( 8x^2y\), x and y are variables and 8 is the coefficient.

Difference Between Monomials, Binomials, and Trinomials

The expressions are classified into monomials, binomials, and trinomials based on the number of terms in them. The terms are parts of an expression separated by mathematical operations like addition and subtraction. Here, we will learn the difference between monomials, binomials, and trinomials. 
 

Monomials Binomials Trinomials 

Expressions with a single term are called monomials.

Binomials are expressions with two terms.

The expressions with three terms are known as trinomials.

Example: 3x, 5xy, \(8x^2y\)

Example: \(2x^2 + y\), \(5x - 4xy\)

Example: \(5x^2 + 4x + 2\), \(3x^2 + 6xy + 9\)

What are Parts of a Monomial?

Monomials consist of one or more variables, their coefficients, and their degree. To better understand a monomial, it can be divided into the following: variable, coefficient, degree, and literal part. 

  • Variable: Variables are letters used to represent unknown values in an expression. The value of expressions depends on the values of variables. 
  • Coefficient: Numerical values that are multiplied by the variables in a term. For example, in the term 7x, 7 is the coefficient. 
  • Degree: The degree of a monomial is the sum of the exponents of all variables. 
  • Literal part: The literal part of a monomial is the part including the variables and their exponents. 

For example: 
 

Monomial  Variables  Degree Literal part \(6x^2y\) x and y \(2 + 1 = 3\) \(x^2y\) \(5x^3y^2z\) x, y, and z  \(3 + 2 + 1 = 6\) \(x^3y^2z\) \(6x^2\) x 2 x2 \(3a^2b\) a and b \(2 + 1 = 3\) \(a^2b\)

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How to Find Monomials?

Algebraic expressions with a single, non-zero term are called monomials. Now let's learn how to identify monomials. The monomials are identified using the following properties: 

  • The monomials should have a single non-zero term.
  • Examples for monomials: 5x2y, 6xy2, etc.
    \(2xy + y\), \(5x^2 + 2x\), \(2x^{1/2} \), are not monomials. 

How to Factorize a Monomial?

To factor a monomial, factor its coefficients and variables separately. When factoring monomials, we separate the coefficients and variables. Let’s learn how to factorize a monomial with an example. 

Example: Factor of the monomial 18x2y

Step 1: Identifying the coefficients and variables

Here, the coefficient is 18

Variables are x2 and y.

Step 2: Factor the coefficient

Prime factorization of 18 = 2 × 3 × 3 .

Step 3: Factoring the variables

Factoring x2 = x × x

Factoring y: y

The complete factorization of 18x2y is 2 × 3 × 3 × x × x × y

What are the Operations on Monomials?

Basic operations like addition, subtraction, multiplication, and division can be performed on monomials. By following simple algebraic rules, we can perform operations on monomials.

Addition of Monomials

If the monomials have the same literal part, we can add them, and the result will be a monomial. We add the coefficients and then keep the literal part the same. 
For example: 5x2y + 15x2y = (5 + 15)x2y =20x2y.

Subtraction of Monomials

Like the addition of monomials, the subtraction of two monomials should have the same literal part. When subtracting two monomials, we first subtract the coefficients and then keep the literal part the same. 
For example: 20x2 - 6x2 = (20 - 6)x2 = 14x2.

Multiplication of Monomials

When multiplying monomials, multiply the coefficients and apply the law of exponents to the variables.

For example: 5x2y × 3xy = (5 × 3)(x2 × x1)(y1 × y1)

= 15x3y2.

Division of Monomials 

Monomials with the same variables can be divided using the quotient law of exponents: (xm / xn = xm - n). First, we divide the coefficients, and then we apply the quotient law of exponents to divide the variables. 

For example: 50x4y2 / 5x2y
(50 / 5) (x4 / x2) (y2 / y) = 10 × (x4 - 2) × (y2 - 1) 
= 10x2y 

Tips and Tricks to Master Monomial

Mastering monomials helps in simplifying algebraic expressions and solving equations efficiently. These tips make calculations faster and more accurate.

  • A monomial is a single term consisting of a number, variable, or product of numbers and variables with whole number exponents.
     
  • Always separate the coefficient (numerical part) from the variable(s) to simplify calculations.
     
  • Apply rules like \(x^a \cdot x^b = x^{a+b} \quad \text{and} \quad (x^a)^b = x^{a \cdot b} \) to handle monomials efficiently.
     
  •  Combine monomials with the same variable and exponent to simplify expressions.
     
  • Multiply and divide monomials by separately handling coefficients and variables using exponent rules.

Common Mistakes and How to Avoid Them in Monomials

Students often make mistakes or misunderstand the properties of monomials. Here are some common mistakes and tips on how to avoid them:

Real-world Applications of Monomial

Monomials are the fundamental concept in algebra and are used in the fields such as physics, finance, biology, and so on. Here are some applications of monomials. 
 

  • In finance, monomials are used in simple interest calculations. The simple interest is calculated using the formula, I = Prt. Here, prt is a monomial.
  • The basic formula in mathematics and physics for calculating distance when time and speed are known is d = st. Here, the expression st is monomial.
  • In biology and medicine, monomial expressions are used to prescribe dosages to patients. If the dosage is 10 mg per kg of the patient’s total body weight w, then the dosage is written as 10w, which is monomial.
  • Monomials are used in calculating electrical power. For example, power \(P\) is calculated as \(P=VI\), where \(V\) is voltage and \(I\) is current. The expression \( VI\) is a monomial.
  • Monomials are used to express concentrations in reactions. For instance, the amount of a substance can be written as \(C×V\), where \(C\) is concentration and \(V\) is volume. The expression \(CV\) is a monomial.

Problem 1

Identify the monomials: 5x²y, ½ y² , 5m + 2n, 2x^(1/2)?

Okay, lets begin

5x2y and ½ y2 are monomials, and 5m + 2n and 2x1/2 are not monomials.

Explanation

The monomials are the expressions with a single term, and the exponent should be a non-negative integer.

5x2y is a monomial as it has only one term, and the exponent is a non-negative integer

½ y2 has only one term, and the exponent is a non-negative integer, so it is a monomial 

The expression 5m + 2n has two terms, so it is not a monomial

The exponent in 2x1/2 is not an integer, so it is not a monomial

Well explained 👍

Problem 2

Multiply the monomials: 5x²y and 2x³y⁴?

Okay, lets begin

The product of \(5x^2y\) and \(2x^3y^4\) is \(10x^5y^5\)

Explanation

To multiply the monomials, we first multiply the coefficients and then multiply the exponents using the law of exponents. 

Multiply the coefficients: \(5 × 2 = 10\)

Multiplying the exponents:\( x^2 × x^3 = x^2 + 3 = x^5 \)

\(y^1 × y^4 =y^1 + 4 = y^5\)

So, the product of 5x2y and \(2x^3y^4\) is \(10x^5y^5\)

Well explained 👍

Problem 3

Find the degree of the monomial 8a²b⁵?

Okay, lets begin

The degree of the monomial in \(8a^2b^5\) is 7

Explanation

The degree of the monomials is the sum of all the exponents of all its variables.

The exponent of a is 2

The exponent of b is 5

So, the degree of \(8a^2b^5\) is 2 + 5 = 7

Well explained 👍

Problem 4

Factorize the monomial: 25x²y⁴?

Okay, lets begin

\(25x^2y^4 = 5 × 5 × x × x × y × y × y × y\)

Explanation

To factorize the monomial, we first break the coefficient and then the variable.

Prime factorization of 25: 5 × 5

Factorizing x2: x × x

Factorizing y4: y × y × y × y

The complete factorization of \(25x^2y^4 : 5 × 5 × x × x × y × y × y × y\)

Well explained 👍

Problem 5

Find the sum of 5x²y and 8x²y

Okay, lets begin

The sum of \(5x^2y\) and \(8x^2y\) is \(13x^2y\)

Explanation

Adding \(5x^2y\) and \(8x^2y\)

Here, both terms have the same variables, so we add the coefficients and keep the same variables.

So, \(5x^2y + 8x^2y = 13x^2y\)

Well explained 👍

FAQs on Monomial

1.What is a monomial?

A monomial is an expression that consists of a single term. It may include one or more variables and with a non-negative integer. For example, 5x2, \(6x^2y^4\), and \(25a^2b\). 

2.Is 2x a monomial?

Yes, 2x is a monomial as it has a single term.

3.Is 7xy a monomial?

Yes, 7xy is a monomial as 7xy is a single term. 

4.What is the degree of a monomial?

The degree of a monomial is the sum of the exponents of all its variables. 

5.Is 5x³ + 5x a monomial?

No, \(5x^3 + 5x\) is not a monomial as it has more than one term, 5x3 and 5x. 

6.What is a monomial and why is it important for my child to learn?

A monomial is a single term consisting of a number, a variable, or a product of numbers and variables with whole number exponents. Learning monomials helps children build a strong foundation in algebra, which is essential for higher-level math and real-life problem solving.

7.How can I help my child identify monomials?

Encourage them to look for single terms that can include numbers, letters, or both. For example, 5x, \(3y^2\), and 7 are all monomials, while \(x+y\) is not because it has two terms.

8.How are monomials used in real life?

Monomials are used in finance (simple interest), physics (distance = speed × time), biology (dosage calculations), engineering (power calculations), and chemistry (concentration × volume).

9.How can I help my child practice monomials at home?

Use practical examples like calculating expenses, distances, or weights. Ask them to write expressions as monomials and simplify them using multiplication and division rules.

10.Are monomials only important for academics?

No, understanding monomials improves logical thinking and problem-solving skills, which are valuable in everyday life and various careers like engineering, finance, and science.

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.