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.
In this lesson, let us learn more about monomials and the ways to factor them.
|2.||Identifying a Monomial|
|3.||Factorizing a Monomial|
|4.||FAQs on Monomial|
Monomial is a type of expression which has only 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 with the variables of the monomial. The degree of a monomial is the sum of the exponents of all the variables. Let’s consider an expression 6xy2. The variables, the coefficient, and the degree of this monomial are shown in the table given below. Observe the table for 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 in a monomial. The exponent of x is 1, and the exponent of y is 2. (2+1=3)||Degree: 3|
Identifying a Monomial
A monomial is very easy to identify. A monomial expression must have a single non-zero term. The exponents of the term must be a non-zero whole number. Further, there should not be any variable in the denominator. Let us look at the following examples to understand more clearly about monomials.
|Polynomial||Is it a monomial?||If no, why?|
|3.||3x2 + y||No||It has two terms: 3x2, and y|
|4.||3x¾||No||The exponent is not a whole number|
|5.||7x||No||The variable is an exponent|
|6.||8x/y||No||The denominator has a variable|
If we observe the third example in the table given above, 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, and an expression that has more than three terms is called a polynomial. In other words, monomials, binomials, and trinomials come under the category of polynomials.
Factorizing a Monomial
Factorizing a monomial is as simple as factorizing a normal number. Consider the number 24. Let’s 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 need to factorize the coefficient and the variables separately. Let’s factorize the monomial, 15y3. The prime factors of the coefficient,15, are 3 and 5. Now, the variable y3 can be factored as y × y × y. Therefore, the complete factorization of the monomial is: 15y3 =3 × 5 × y × y × y.
Important Notes on Monomials
Observe the following points which help in understanding the results of the arithmetic operations on a monomial.
- A single term expression in which 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.
Solved Examples on Monomials
Example 1: Choose the monomials from the following expressions: (a) x2 (b) 3 (c) x+5y (d) 8/x
The expressions x2 and 3 are monomials because they have a single non-zero term and have exponents as whole numbers. The expression x+5y is not a monomial because it has two terms. The expression 8/x is also not a monomial because it has a variable as its denominator. Therefore, among the given expressions, x2 and 3 are considered to be monomials.
Example 2: Factor the monomial expression 10xy completely. Also, draw its factor tree.
In the given monomial 10xy, the prime factors of the coefficient 10 are 2 and 5. The variable part xy can be split as x × y.
Therefore, the complete factorization of the monomial is: 10xy = 2 × 5 × x × y.
Example 3: Jenna came across an expression 12y/x. She asked Emma if it is a monomial or not. Can you help her?
The expression has a single non-zero term, but the denominator of the expression is a variable. So, it is not a monomial. Therefore, the expression 12y/x is not a monomial.
Example 4: Jacob and Sam wrote the factors of the monomial, 27x4 as follows. Jacob: 3x2, 9x2 , Sam: 3x, 3x2. Which of the students factored 27x4 correctly?
Jacob factorized 27x4 as: 3x2, 9x2 , while, Sam factorized it as 3x, 3x2. If we multiply the factors listed by Jacob, we get (3x2) × (9x2 ) = 27x4. However, when we multiply the factors listed by Sam, we get (3x2) × (3x ) = 9x3. Therefore, Jacob listed the factors of 27x4 correctly.
FAQs on Monomials
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 has its exponent as a whole number. Further, the term does not have a variable associated with the denominator.
Can a Monomial be a Polynomial?
Yes, a monomial is a type of polynomial, having a single non-zero term. In other words, a polynomial containing a single term is called a monomial.
What is a Monomial?
A monomial is a polynomial with only one term. It is basically composed of variables and a coefficient. In exceptional cases, a monomial can be made up of more than one variable. A few of the examples of monomials are 5x, 2y3, 7xy, x5.
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 the 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. (1+3=4).
What is a Constant Monomial?
A constant monomial is a monomial 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 they have, and all these come under the category of polynomials. 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.