A monomial is a polynomial, which has only one term. A monomial is an algebraic expression with single term, but can have multiple variables and higher degree too. For example, 9x³yz is a single term, where 9 is the coefficient, x,y,z are the variables and 3 is the degree of monomial. Similar to polynomial, a monomial involves the operation of addition, subtraction, multiplication, and non-negative integer exponents of variables. In this article, we are going to learn the monomial definition, different arithmetic operations performed on monomials and examples in detail.

What is a Monomial?

A monomial is a type of polynomial, which is an algebraic expression having only a non-zero single term. Monomial consists of only a single term which makes it easy to do the operation of addition, subtraction and multiplication. It consists of either only one variable or one coefficient or product of a variable and a coefficient with exponents as whole numbers, which represent only one term, unlike binomial and trinomial, which consist of two and three terms respectively. It cannot have a variable in the denominator.

Examples of Monomial

Let us consider some of the variables and examples:

• p – One variable and degree is one.
• 5p² – with 5 as coefficient and degree as two.
• p³q – with two variables and degree as 4(3+1).
• -6ty – two variable t and y and a coefficient -6
• Let us consider x³+3x²+4x+12 is a polynomial, where x³,3x²,4x and 12 are single terms and called monomials.

Parts of Monomial Expression

The different parts present in the monomial expression are:

• Variable: The letters present in the monomial expression.
• Coefficient: The number which is multiplied by the variable in the expression
• Degree: The sum of the exponents present in the expression
• Literal part: The alphabets which are present along with the exponent value in the expression

Example: 4xy² is a monomial expression.

Here, 

the coefficient is 4

Variables are x and y

The degree of the monomial expression = 1+2 = 3

The literal part is xy²

Like, 4x is a monomial example, as it denotes a single term. In the same way, 23, 4x², 5xy, etc.are examples but 23+x, 4x^y, 5xy^-2 are not, as they don’t fulfil the conditions.

It is a product of powers of variables with non- negative integer exponents, such that, if there is a single variable x, then it has either a 1 or a power of xⁿ of x, with n as positive integer and if for the product of multiple variables such that XYZ, then the monomials can be given in the form of x^a y^b z^c where a,b,c are non-negative integers.

Now, in terms of the coefficient, it is defined as the term with a non zero coefficient. The degree of a monomial is the sum of the exponents of all the included variables which forms monomials. For example, xyz² have three degrees, 1,1 and 2. Therefore, the degree of xyz² is 1+1+2 = 4.

Degree of Monomial

The degree of a monomial expression or the monomial degree can be found by adding the exponents of the variables in the expression. While calculating the monomial degree, it includes the exponent values of the variables and it also includes the implicit exponent of 1 for the variables, which usually does not appear in the expression.

For example, 2xy³. In this, the exponent value of 1 is not visible in the expression. Thus, the degree of the expression is 1+3 = 4. In case, the monomial expression is a constant value. The degree of the non-zero constant is given as 0. 

The degree of the monomial expression is also called the order of the monomial.

Factorization of Monomial

Like factoring a number, the monomial expression can also be factored. For example, the factorization of 15 is 3×5. The monomial expression can be expressed in the same way. Now, consider a monomial expression, 24a³. First, factor the coefficient of the variable, (i.e) 24. The number 24 is factored as 2×2×2×3. Similarly, a³ is factored as a×a×a.

Therefore, the factorization of the monomial 24a³ is 2×2×2×3×a×a×a.

Operations on Monomial

The arithmetic operations which are performed on the monomial expression are addition, subtraction, multiplication and division.

• Addition of two monomials
• Subtraction of two monomials
• Multiplication of two monomials
• Division of two monomials

Addition of Two Monomials

The addition of two monomials with the same literal part will result in a monomial expression

For example, the addition of 4ab + 6ab is 10 ab.

Subtraction of Two Monomials

The subtraction of two monomials with a similar literal part will result in a monomial expression

For example, the subtraction of 10xyz – 3xyz is 7xyz.

Multiplication of Two Monomials

The multiplication of two monomials will also result in monomial

For example, the product of 3x²y and 4z is 12x²yz

While multiplying two monomials with the same variables, then add the exponent value of the variables.

For example, the product of a³ and a⁴ is given as

(a³)(a⁴) = a³⁺⁴ = a⁷.

Division of Two Monomials

While dividing two monomials with the same variables, subtract the exponent value of the variables.

For example, the division of a⁹ by a³ is given as

(a⁹) / (a³) = a^9-3 = a⁶.

Difference Between Monomial, Binomial and Trinomial

Now hopefully, we have got the basic difference between Monomial, Binomial and Trinomial. Let us solve some problems based on monomial.

Important Facts of Monomials

• The multiplication of two monomial will also result in the monomial.
• The sum or difference of two monomials might not result in a monomial.
• An expression having a single term with a negative exponent cannot be considered as a monomial. (i.e) A monomial cannot have variables with negative exponents.

Related Articles

• Binomial
• Degree of a Polynomial
• Factorization of Polynomials
• Multiplying Polynomials

Solved Problems on Monomials

Example 1:

Identify which of the following is a Monomial.

1. 3ab
2. 4b+c
3. 6x²+2y
4. a+b+c²

Solution:

3ab is a Monomial

Whereas 4b+c and 6x²+2y are binomials and a+b+c² is a trinomial.

And all of these equations are called polynomial.

Example 2: 

Find the factorization of the monomial 10y³.

Solution:

Given monomial: 10y³.

First, factorize the coefficient of the variable, y. (i.e.)10.

Hence, 10 can be factorized as 2×5.

y³ can be factorized as y × y × y.

Therefore, the factorization of the monomial 10y³ is 2×5×y×y×y.

Monomial Practice Questions

1. Categorize the following expressions into monomials, binomials and trinomials.

(a) 2x+3y

(b) 4m³

(c) 2x²+3m-2

(d) -2y³

2. Factorize the monomial expression 64x²y.

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