Monomial

Introduction

• A monomial is the product of positive integer powers of variables.

• it does not include any negative number and fraction exponent.

• If the monomial is a fraction it has no variable in its denominator.

• A monomial variable is a character it contains.

• Coefficients are numbers multiplied by monomial variables.

• The degree of the monomial is the sum of the exponents of all variables.

In this tutorial, we will discuss Monomial and its parts.

What is a Monomial?

• A monomial is a component of a polynomial and is called a "term" when it is part of a larger polynomial.

• In other words, every term in a polynomial is a monomial.

• A monomial in Math is a type of polynomial that has only one single term.

• For example, 6s + 7s + 8s is a monomial because when we add similar terms it will obtain the result as 21s. Furthermore, 8y, 9x²y, 3xy, etc are monomials because each of these expressions includes only one term.

• A polynomial consists of the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. In monomials, the exponent should be an integer.

Examples of Monomials

There are various examples of monomials which are discussed below.

• 2x² y³z

• 5x³ y

• 10z³

• a – Here, the variable is one and degree is also one.

• 7x⁴– Here, the coefficient is 7 and the degree is 4

• x³y – Here, x and y are two variables and the degree is (3+1)=4.

• -9xy – Here, x and y are two variables and the coefficient is - 9.

Parts of Monomial Expression

The various part present in the monomial are-

Variable − variable is the letters which presents in monomial whose value is not fixed.

Adding degree is expressed as the sum of exponents present in the monomial expression.

Coefficient − the coefficient is the number that is multiplied by the variable in the monomial expression.

Literature Part − this is the part present with the exponent value in form of alphabets in the monomial expression.

Let us an example of a monomial expression.

Ex- 9xy²

Here in this example as we can see that 9 is a coefficient of monomial expression.

X and y are the variables present in the monomial expression.

As we see the degree of this monomial expression is calculated by 1+2 = 3.

Now look at some more examples of monomial expressions - 8xy, 24ab^2., 22a these all the monomial, let we see some expression like 2+x, 6+y, 9+xy these all are not an example of monomial because they not fulfill the above condition of monomial.

Degree of Monomial

Now it's time to discuss some points about the degree of a monomial, as we see the degrees expressed as the sum of exponents present in the monomial expression. always remember that the degree of the nonzero constant is always zero.

For ex- 9x² y³ in this expression, the degree is 2+3 = 5 here monomial value is constant. Order- the second name of the degree of a monomial is order.

Factorization of Monomial

Factorization of monomial expression is similar to as we factorized the number for ex :- 12b⁴ is first factored in the coefficient as 2×2×3 and similarily b⁴ is factorized as b×b×b×b.

Operations on Monomial

The various arithmetic operation which we use in the monomial expression is, addition, subtraction, multiplication, and division.

For example

Two monomials in Addition.

Two monomials in Subtraction.

Two monomials in Multiplication.

Two monomials in Division.

Addition of Two Monomials

By adding the two monomial which has similar literal part will result following monomial expression.

For example, the addition of 8xy+ 11xy is 19xy.

Subtraction of Two Monomials

By subtracting the two monomial which has similar literal part will result following monomial expression.

For example, the subtraction of 12abc – 8abc is 4abc.

Multiplication of Two Monomials

By multiplying monomial with monomials will also result in a monomial,

For example,

the product of 6x² y and 5z is 30x² yz

When we multiplying two monomials with the same variable , then we add their exponent value.

For example,

the product of x⁶and x⁴is given as

(x⁶)(x⁴) = x⁶⁺⁴= x¹⁰

Division of Two Monomials

When we divide two monomials with the same variables , then we subtract their exponent value.

For example, the division of x⁸ by x² is given as

$$\mathrm{\frac{x^8}{x^2} =x^{8-2}=x^6}$$

Difference Between Monomial, Binomial, and Trinomial

Monomial	Binomial	Trinomial
A monomial is an expression in algebra that contains one term. As we can see in the examples there is no need for any mathematical operators.	An expression that contains two different terms is called a binomial; for example, x + y, m – 5, mn + 4m, and a 2 – b 2 are binomials. As you can see here we are using the + or – sign because there are two terms connected these mathematical operators.	An expression that contains three terms is called a trinomial. Now it is already clear here we use two operators ( + or – sign) to connect three terms.
Examples- 2a, 4b, 6c, 2x2, 7abc, etc., are the monomials.	Example- 2x2 + y, 10p + 7q2, a - b, 2x2 y2+ 9, are the binomials.	Example- 2x2 + y - z, r - 10p + 7q2, a + b - c, 2x2 y2 + 9 + z, are the trinomials.

Solved Problems

Example1. Select the monomial in the following question.

• 4ab

• 9b+c

• 5x²+2y

• a+b+c²

Solution: 4ab is a monomial.

9b+c and 5x²+2y are binomials and a+b+c² is a trinomial.

And all of them is polynomials.

Example2. Find the factor of the following monomial 8a².

Solution- Given the monomial is 8a²

First we factorize the coefficient of the variable a . (i.e.)8

Here, 8 is factorized as 2×2×2

a² can be factorized a×a.

Therefore, the factorization of the monomial 8a² is 2 × 2 × 2 × a × a.

Example3. Is 7ab + ab monomial or binomial?

Solution: The given expression: (7ab + ab) = 8ab, which is an expression containing only one term, therefore it is a monomial.

Example4. Factorize the monomial, 15y⁴.

Solution: 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 y⁴ can be factorized as = y × y × y × y.

Therefore, the complete factorization of the monomial is 15y⁴ = 3 × 5 × y × y × y × y.

Example 5. Simplify the following equation (x⁵ y²)(x³ y²)

Solution: (x⁵ x³)(y² y²)

= x⁵⁺³×y²⁺²

= x⁸ y⁴

Conclusion

A polynomial with a single term is called a monomial. In algebra, a monomial is an expression with a single term with variables and coefficients. For example, 4xy is a monomial because it has two variables and one coefficient. A monomial is a component of a polynomial and is called a "term" when it is part of a larger polynomial. In other words, every term in a polynomial is a monomial.

Parts of monomial expression are

Variables

Coefficient

Degree

Literal part

• When we multiply two monomials with the same variables, then we can add their exponent value of the variables.

• When we divide two monomials with the same variables, then we can subtract the exponent value of the variables.

• Monomial is an expression with a single term. Binomial is an expression with two non-zero terms. Trinomial is an expression with three non-zero terms.

FAQs

1.What is a monomial?

A monomial is an expression in algebra that contains one term, like 3xy. It includes numbers, whole numbers and variables that are multiplied together, and variables that are multiplied together.

2.How to find the degree of monomials?

The degree of the monomial is the sum of the exponents present in the monomial expression.

3.Is the 5ˣ term a monomial?

No, this is not a monomial because the exponent is not an integer.

4.How to check whether the given expression is monomial or not?

The expression should be of one non-zero term.

The exponent should be an integer.

There should not be any variable in the denominator of expression.

5.What is a constant monomial? Give an example.

A polynomial with a single non-zero constant term.

For example,$\mathrm{25, 5, \frac{7}{8} \:etc.}$