A **monomial** is a polynomial, which has only one term. As we know, the polynomials are the equations or algebraic expressions which consists of variables and coefficients and has one or more terms in it. Each term of the polynomial is a monomial. A polynomial involves the operation of addition, subtraction, multiplication, and non-negative integer exponents of variables.

**Table of Contents:**

## Monomial Definition

A monomial is a type of polynomial, like, binomial and trinomial, which is an algebraic expression having only a single term, which is a non-zero. It 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.

**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

For example, 4xy^{2} 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 xy^{2}

Like, 4x is a monomial example, as it denotes a single term. In the same way, 23, 4x^{2}, 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 x^{n} 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, xyz^{2 }have three degrees, 1,1 and 2. Therefore, the degree of xyz^{2 } is 1+1+2 = 4.

## Monomial Degree

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, 2xy^{3}. 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.

## Monomial Operations

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

**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 3x^{2}y and 4z is 12x^{2}yz

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

For example, the product of a^{3} and a^{4 }is given as

(a^{3})(a^{4}) = a^{3+4} = a^{7}.

**Division of Two Monomials:**

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

For example, the division of a^{9} by a^{3 }is given as

(a^{9}) / (a^{3}) = a^{9-3} = a^{6}.

## Monomial Examples

Let us consider some of the variables and examples:

- p – One variable and degree is one.
- 5p
^{2}– with 5 as coefficient and degree as two. - p
^{3}q – with two variables and degree as 4(3+1). - -6ty – two variable t and y and a coefficient -6
- Let us consider x
^{3}+3x^{2}+4x+12 is a polynomial, where x^{3},3x^{2},4x and 12 are the single terms and called as monomials.

### Binomial

A binomial is a polynomial or algebraic expression, which has a maximum of two non-zero terms. It consists of only two variables.

Example are: 2x^{2 }+ y, 10p + 7q^{2}, a + b, 2x^{2}y^{2 }+ 9, are all binomials having two variables.

### Trinomial

A trinomial is a polynomial or algebraic expression, which has a maximum of three non-zero terms. It consists of only three variables.

Example are: 2x^{2 }+ y + z, r + 10p + 7q^{2}, a + b + c, 2x^{2}y^{2 }+ 9 + z, are all trinomials having three variables.

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

### Monomial Problems

**Question: **Identity which of the following is a Monomial.

- 3ab
- 4b+c
- 6x
^{2}+2y - a+b+c
^{2}

**Answer:** 3ab is a Monomial

Whereas 4b+c and 6x^{2}+2y are binomials and a+b+c^{2 }is a trinomial.

And all of these equations are called as a polynomial.

## Frequently Asked Questions on Monomials

### Define monomial, binomial and trinomial

A monomial is an expression with only one term. Example. 3x.

A binomial is an expression with two terms. Example 2x+3y

A trinomial is an expression with three terms. Example x+2y+3z

### What is meant by the degree of a monomial?

The degree of a monomial is defined as the sum of the exponents of the variables present in the monomial term.

### What are the different arithmetic operations performed using monomials?

The different arithmetic operations performed using monomials are addition, subtraction, multiplication and division.

### Can we get a monomial term while adding two monomials?

If two monomials with the same literal parts are added, the sum should be a monomial. In case, the addition of two monomials with the different literal part, the result should be a binomial.

### How to identify the monomial expression?

The monomial expression should not have an addition or subtraction operator. A monomial can be a constant term or else, the variables with coefficients and exponents.

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