Definition, Degree | Monomial Examples - Cuemath

Monomial

In algebra, a monomial is an expression that has a single term, with variables and a coefficient. For example, 2xy is a monomial since it is a single term, has two variables, and one coefficient. 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.

1.	What is Monomial?
2.	Monomial Binomial Trinomial
3.	Degree of a Monomial
4.	Factoring Monomials
5.	FAQs on Monomial

What is Monomial?

Monomial is defined as an expression that has a single non-zero term. It consists of different parts like the variable, the coefficient, and its degree. The variables in a monomial are the letters present in it. The coefficients are the 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. Let us consider an expression 6xy2. The variables, the coefficient, and the degree of this monomial are shown in the table given below. Observe the table to learn the various parts of the monomial 6xy2.

The variables are the letters present in a monomial.	Variables: x, y
The coefficient is the number that is multiplied by the variables.	Coefficient: 6
The degree is the sum of the exponents of the variables in a monomial. The exponent of x is 1, and the exponent of y is 2, so the degree is 2 + 1 = 3.	Degree: 3

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.

Let us look at the following examples to identify monomials.

Expression	Is it a monomial?	If not, why?
3x2y	Yes	-
3y/2	Yes	-
3x2 + y	No	It has two terms: 3x2, and y
3x¾	No	The exponent of the variable is not an integer
7x	No	The variable is an exponent
8x/y	No	The denominator has a variable

Monomial Binomial Trinomial

If we observe the third example in the table given above, that is, 3x2 + y, we see that it has 2 terms. An expression having two terms is called a binomial. Similarly, an expression having three terms is called a trinomial. For example, 4x2 + 2y + 6z is a trinomial. It is important to note that monomial, binomial, and trinomial are all types of polynomials. Look at the image given below to understand the difference between monomial, binomial, and trinomial.

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 abc2 is 4. The exponent of the variable 'a' is 1, the exponent of variable 'b' is 1, the exponent of variable 'c' is 2. Adding all these exponents, we get, 1 + 1 + 2 = 4. Let us learn how to find the degree of a monomial with another example.

Example: Find the degree of the monomial: -4xy.

In the given term, the coefficient is -4, and x and y are the variables. The exponent of the variable x is 1. The exponent of the variable y is 1. Therefore, the degree of the monomial is the sum of these exponents, that is, 1 + 1 = 2.

Factoring Monomials

While factoring monomial, we always factor coefficient and variables separately. Factorizing a monomial is as simple as factorizing a whole number. Consider the number 24. Let us see the factors of this number. The number 24 can be split into its factors as shown in the following factor tree:

In the same manner, we can factorize a monomial. We just need to remember that we always factorize the coefficient and the variables separately.

Example: Factorize the monomial, 15y3.

In the given monomial, 15 is the coefficient and y3 is the variable.

• The prime factors of the coefficient,15, are 3 and 5.
• The variable y3 can be factored in as y × y × y.
• Therefore, the complete factorization of the monomial is 15y3 = 3 × 5 × y × y × y.

Tips and Tricks on Monomials

Observe the following points which help in understanding the results of the arithmetic operations on a monomial.

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

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