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.

What Is a Monomial?

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.

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Step by step guide to monomial

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 for Monomial

Factor each of the following monomial.

  1. \(\color{blue}{10x^2}\)
  2. \(\color{blue}{8 x^2 y^2}\)
  3. \(\color{blue}{18 x y}\)
This image has an empty alt attribute; its file name is answers.png
  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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