What is Monomial?

A monomial is a single-term algebraic expression. It can be a constant, a variable, or a product of numbers and variables where the variables have non-negative integer exponents. Monomials are the simplest forms of polynomials and are essential in algebraic expressions and equations.

In general, a monomial can be written as:

[Tex]a \cdot x_1^{k_1} \cdot x_2^{k_2} \cdots x_n^{k_n}[/Tex]

​​Where:

• a is a constant (also called the coefficient),
• x₁, x₂, . . ., x_n​ are variables,
• k₁, k₂, . . . , k_n​ are non-negative integers (the exponents).

Key points:

• A monomial always contains a single term.
• Variables must have non-negative integer exponents.
• No addition, subtraction, or division of terms or variables.

Parts of Monomials

The various parts of the monomial are covered in the below table with examples.

Example of Monomial

"Mono" refers to one. When polynomials are classified on the basis of the number of terms, the polynomials with only a single term are called monomials. Some of the examples of monomials are:

Constant Monomials

• 5
• -3.14
• 1000

Monomials with a Single Variable

• 2x
• -0.5y
• 4x²
• 0.25y³

Monomials with Multiple Variable

• 6xy
• -2.5yz²
• 3x²y³

Degree of Monomial

The sum of the exponents of the variables in a monomial gives us the degree of monomial.

This can be well understood by the following example.

The exponents of all the variables are added to determine a monomial's degree. It is always an integer that is not zero. For instance, the monomial xyz³ has a degree of 5.

• Variable 'x' has an exponent of 1,
• Variable 'y' has an exponent of 1, and
• Variable 'z' has an exponent of 3.

The result of adding all these exponents is 1 + 1 + 3 = 5.

Example 1: Find the degree of monomial -3x³y³.

Solution:

Here the exponent of x is 3

Exponent of y is 3

So degree = 3 + 3 = 6

Example 2: Find the degree of the monomial 9xy³.

Solution:

Degree of a monomial is given by the sum of exponents of the variables in a monomial.

Exponent of x = 1, Exponent of y = 3

Degree = 1 + 3 = 4

Identifying a Monomial

Let's apply the properties of a monomial in order to identify a monomial in the below examples:

Let's consider an example for better understanding.

Example: Identify if the following are monomial or not.

• x + 2y
• 7x²y

Solution:

• For x + 2y, Contains two terms that is x and 2y so it is not a monomial.

• For 7x²y, Contains a single term so it is a monomial.

Monomial Binomial and Trinomial

A monomial expression has, only one term. For example, 3xy is a monomial. A binary expression is one which has two terms. For example, 3x+4y, 4xy+6z is a binomial. Similar to this, a trinomial is an expression with three terms. For instance, a trinomial is 4x² + 2y + 6z or 5x + 7xy + 9z .

Lets look into the table below for more clarification between the three terms:

Note: Monomial is a type of polynomial. All monomial are polynomial but all polynomial aren't monomial.

Factors of Monomials

We usually factor coefficient and variables independently while factoring monomial. A monomial can be factored just as easily as a whole number.

Example: Factorize the monomial, 20y⁴.

In the given monomial, 20 is the coefficient and y⁴ is the variable.

• The prime factors of the coefficient, 20; are 2, 2, and 5.
• The variable y⁴ can be factored in as y × y × y × y.

Therefore, the complete factorization of the monomial is 20y⁴ = 2 × 2 × 5 × y × y × y × y.

Read More about Factoring Polynomial.

Operation on Monomials

Operations on monomials involve:

• Addition
• Subtraction
• Multiplication
• Division

Let's discuss these operations in detail as follows:

Addition of Monomials

To add two or more monomials that are like terms, add the coefficients; keep the variables and exponents on the variables the same.

Example: Add 10xy² and -9xy².

Solution:

In both monomials 10xy² and -9xy²; xy² is common.

Thus, 10xy² + (-9xy²) = (10 + (-9))xy² = xy²

Subtraction of Monomials

To subtract two or more monomials that are like terms, subtract the coefficients; keep the variables and exponents on the variables the same.

Example: Subtract 10xy² and -9xy²

Solution:

Subtract the coefficient only

(10 - (-9))xy² = 19xy²

Monomial Multiplication

To multiply a monomial by a monomial we get a monomial. The coefficients of the monomials are multiplied together and then the variables are multiplied.

Example: Find the product of two monomials 2x and 2y.

Solution:

Product of two monomial = 2x*2y = 4 xy

Monomial Division

To divide a monomial by a monomial, divide the coefficients and divide the variables with like bases by subtracting their exponents. To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

Example: Divide 4xy by 2x.

Solution:

Division of 4xy by 2x

= 4xy/2x = 2y

Read More,

• Types of Polynomials
• Polynomial Formula
• Multiplying Polynomials
• Dividing Polynomials 

Solved Problems on Monomials

Problem 1: Choose the monomials from the following expressions:

(a) x³
(b) 4 - x

Solution:

(a) x³ is a monomial as it has a single term.

(b) 4 -x isn't a monomial as it has two term.

Problem 2: Factorize the monomial expression: 8xy.

Solution:

In 8xy, the prime factors of coefficient 8 are 2 and 4. The variable part 'xy' can be split as x × y.

Therefore, the complete factorization of the monomial is 8xy = 2 × 4 × x × y.

Problem 3: Is 10y/x a monomial expression? Justify your answer.

Solution:

The expression has a single non-zero term, but the denominator of the expression is a variable. Therefore, the expression 10y/x is not a monomial.
 

Problem 4: Find the degree of monomial 48 xy³.

Solution:

Degree of monomial = sum of exponents of all variable

Thus, Degree of monomial = 1 + 3 = 4

Practice Problems on Monomials

Problem 1: Multiply the monomials: 4x³ and 2x².

Problem 2: Simplify the expression: 3a⁴b² · 5a²b³.

Problem 3: Divide the monomials: 8x⁵/4x².

Problem 4: Add the monomials: 2x³ + 3x³.

Problem 5: Subtract the monomials: 7y⁴ - 2y⁴.

Problem 6: Find the product of the monomials: -2x⁴ · -3x².

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Article Tags :

• Mathematics
• School Learning
• Algebra
• maths-polynomial
• Maths
• Polynomials

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