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.

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.

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.

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.

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.

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.

☛ Related Topics

Check these interesting articles related to monomials in math.

• Monomial Calculator
• Dividing Polynomials
• Dividing Monomials

What is a Monomial in Math?

Monomial is an expression that has a single non-zero term. Monomials can be numbers, variables, or numbers multiplied with variables. For example, 2, ab, and 42xy are examples of a monomial. A few other examples of monomials are 5x, 2y³, 7xy, x⁵.

How to Factor a Monomial?

In order to factorize a monomial, the coefficient and the variables need to be factorized separately. For example, to factorize 26y², first, we factorize 26 as 2 × 13. Then, y² can be factorized as y × y. So, 26y² can be factorized as 2 × 13 × y × y.

Is x² a Monomial?

Yes, x² is a monomial because x² is a single non-zero term and its exponent is a whole number. Further, the term does not have a variable associated with the denominator.

Is XYZ a Monomial?

Yes, the term XYZ is a monomial. Even with three variables, it is a single term, thus, it can be called a monomial. In other words, a monomial can be made up of more than one variable. Here x, y, z are its variables.

What is the Degree of a Monomial?

The degree of a monomial is the sum of the exponents of all its variables. For example, in the monomial 2xy³, the exponent of x is 1, and the exponent of y is 3. So, the degree of this monomial is 4, i.e. 1 + 3 = 4.

What is a Constant Monomial?

A constant monomial is a term with only a constant number. There is no variable in a constant monomial. The terms 5, 22/7, 1/2, 11 are all examples of constant monomials.

What is the Difference Between Monomial, Binomial, Trinomial?

Monomials, binomials, and trinomials are all named according to the number of terms that they have. An expression with a single term is a monomial, for example, 4x, 5x², 7x⁴. An expression having two terms is called a binomial, like, 11x + 2xy, or, 13y + x³. An expression having three terms is called a trinomial, like, 4x + x² + 9x³.

How to Divide a Polynomial by a Monomial?

In order to divide a polynomial by a monomial, we divide each term of the given polynomial separately by the monomial. Let us understand this with an example using the following steps:

• Divide the polynomial (8a³ + 40a² + 20a) by the monomial 4a.
• Divide each term of the polynomial by the monomial. This means, (8a³)/4a + (40a²)/4a + (20a)/4a.
• After simplifying each term, we will get, 2a² + 10a + 5.

How to Find the Greatest Common Factor of Monomials?

In order to find the greatest common factor of two monomials, we need to do the prime factorization of both the monomials. For example, the two monomials are 9a²b and 36abc. After the prime factorization of 9a²b, we get, 3 × 3 × a × a × b. The prime factorization of 36abc is 2 × 2 × 3 × 3 × a × b × c. Now, if we find the common factors among these, we get, 3 × 3 × a × b, which can be written as 9ab. Therefore, the greatest common factor of the two monomials is 9ab.

How to Identify 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 whole numbers.
• There should not be any variable in the denominator.

How to Multiply a Polynomial by a Monomial?

In order to multiply a polynomial by a monomial, we need to multiply each term of the polynomial with the given monomial. Let us understand this with an example using the following steps:

Multiply the monomial (3a²) with the polynomial (2a + 5ab - 4).

• Step 1: Multiply the monomial with the first term of the polynomial. This means, 3a² × 2a = 6a³.
• Step 2: Again, multiply it with the second term of the polynomial. This means, 3a² × 5ab = 15a³b.
• Step 3: Now, multiply it with the third term of the polynomial. This means, 3a² × -4 = -12a².
• Step 4: Write all the terms together with their corresponding signs, to get the product, that is, 6a³ + 15a³b -12a².