# What Is a Monomial?

In algebra, a monomial is an expression that has a single term, with variables and a coefficient. In this guide, you will learn more about the definition of monomial and its factoring.

Monomials are the building blocks of polynomials and are called terms when they are a part of larger polynomials. In other words, each term in a polynomial is a monomial.

## Step by step guide tomonomial

A monomial is defined as a phrase that has a single non-zero term. It consists of different parts such as variable, coefficient, and degree. The variables in a monomial are the letters in it. Coefficients are numbers that are multiplied by the variables of the monomial. The degree of a monomial is the sum of the exponents of all the variables.

### How to find a monomial?

A monomial can be easily identified with the help of the following properties:

• A monomial expression must have a single non-zero term.
• The exponents of the variables must be non-negative integers.
• There should not be any variable in the denominator.

### Degree of a monomial

The degree of a monomial is the sum of the exponents of all the variables. It is always a non-negative integer. For example, the degree of the monomial $$abc^2$$ is $$4$$. The exponent of the variable $$a$$ is $$1$$, the exponent of variable $$b$$ is $$1$$, and the exponent of variable $$c$$ is $$2$$. Adding all these exponents, we get, $$1+1+2=4$$.

### Factoring monomials

When factoring monomial, we always factor coefficient and variables separately. Factoring a monomial is as simple as factoring a whole number. Consider the number $$24$$. Let’s see the factors of this number. As shown in the following factor tree, the number $$24$$ can be divided into its factors:

In the same way, we can factorize a monomial. We just have to remember that we always factorize coefficients and variables separately.

Example: Factorize the monomial, $$15y^3$$.

Solution:

In the given monomial, $$15$$ is the coefficient, and $$y^3$$ is the variable.

• The prime factors of the coefficient,$$15$$, are $$3$$ and $$5$$.
• The variable $$y^3$$ can be factored in as $$y×y× y$$.
• Therefore, the complete factorization of the monomial is $$15y^3 = 3 × 5 × y × y × y$$.

#### Tips and Tricks on Monomials:

• A single term expression whose the exponent is negative or has a variable in it is not a monomial.
• The product of two monomials is always a monomial.
• The sum or difference of two monomials might not be a monomial.

### Monomial – Example 1:

Is $$\frac{14z}{x}$$ a monomial expression?

Solution:

The expression has a single non-zero term, but the denominator of the expression is a variable. Therefore, the expression $$\frac{14z}{x}$$ is not a monomial.

## Exercises forMonomial

### Factor each of the following monomial.

1. $$\color{blue}{10x^2}$$
2. $$\color{blue}{8 x^2 y^2}$$
3. $$\color{blue}{18 x y}$$
1. $$\color{blue}{2. 5. x. x}$$
2. $$\color{blue}{2. 2. 2. x. x. y. y}$$
3. $$\color{blue}{2. 3. 3. x. y}$$

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