A monomial is defined as an algebraic expression which has exactly one term. Its general form can be written as $Ax^{m}y^{n}$ where, A is the coefficient, x, y are the variables and m, n are its exponents.

Monomials can include numbers, numbers and variables combined together by multiplication or variables combined by multiplication.

An algebraic expression which contains only one term is called a monomial.

For example, $-7z, 8xy, a^{2}b^{3}$ are some examples of monomials.

## What is a Monomial?

An expression of the form $ax^{n}$ is called a monomial in 'x'.
Here,
'a' is a known number, called the "coefficient of x"
'x' is a variable
'n' is a non-negative integer, called the "degree of the monomial".

'mono' means "one". A monomial contains only one term and it has no positive or negative sign.

### Examples of Monomials

2x is a monomial with 2 as a coefficient, x as a variable and 1 as a degree.
$5y^{2}$ is a monomial with 5 as coefficient, y as variable and 2 as degree.

### Characteristics of monomials:

• A monomial can be a constant or a variable or a combination of both.
• It must contain non negative whole numbers as exponents for the variable.
• The coefficient must be an integer or a fraction in the simplest form.
• The only operation between the coefficient and variables is multiplication.
• The coefficient of the monomial is the number that is mostly written in front of the variable.
• x = x¹
• x° = 1, as any number raised to the power 0 is always 1.

Monomial is a polynomial with only one term.

### Coefficient of a Term:

Any factor or a group of factors of a product are known as the coefficient of the remaining factors.
For example, in the product 12ab,

• 12 is numeral coefficient of ab
• 12a is the coefficient of b
• ab is the coefficient of 12 and so on

When the coefficient is unity(1), it is usually omitted.
For example, 1x is written as x

## Parts of a Monomial

The literal and exponent part forms the literal coefficient of the monomial.

Example: The literal coefficient of $5 x^{2} y^{3}$ is $x^{2} y^{3}$

The sum of all the exponents of the variable of a monomial can be defined as its degree.

Example: The degree of the monomial $24 x^{3} y^{3} z$ can be taken as the sum of the corresponding exponents = 3 + 3 + 1=7

## Comparison of Monomials

If the literal coefficient of two or more monomials is same, then they are said to be similar.

Example:
5mn and 6mn are similar since both have same literal coefficient mn.

$4 x^{2} y$ and $4 x y^{2}$ are different as they don’t have same literal coefficient.

## Power of Monomials

Power raises every element of the monomial to its exponent power.
$(a y^{m})^{n}$ = $a^{n} y^{(mn)}$

Example: $(5 x^{2}) ^{3}$ = $5^{3}(x^{2}) ^3$= $125 x^{6}$

Addition is only possible if the literal coefficients are the same.
The operation performed is

$A x^{b} + B x^{b}$= $(A+B) x^{b}$
• The sum of two or more monomial is again a monomial.
• The sum of non-similar monomials is a polynomial.

For example,
$7 x^{2} y + 3 x^{2} y$ = $(7+3) x^{2} y$ = $10 x^{2} y$
$a b + 6 x^{2} y^{3}$ (Here, the sum of two monomial is a binomial.)

Given below are some of the examples that explains how to add monomials.

Example 1:

Adding monomials: $3 x y^{4} + 8 x y^{4}$

Solution:

Given $3 x y^{4} + 8 x y^{4}$

Step 1: We add like terms $(3 + 8)$ and variables $x y^{4}$ remain same
$(3+8) x y^{4}$
Step 2: After combining like terms we get $11 x y^{4}$
Answer: $11 x y^{4}$

Example 2:

Adding monomials: $9 p + 14 p$

Solution:

Given $9 p + 14 p$

Step 1: We add like terms $(9 + 14)$ and variable $p$ remain same
$(9 + 14) p$
Step 2: After combining like terms we get $23 p$
Answer: $23 p$

## Subtracting Monomials

Subtraction is only possible if the literal coefficients are the same.
The operation performed is

$A x^{b} - B x^{b}$ = $(A-B) x^{b}$
• The difference of two or more monomials is again a monomial.
• The difference of non-similar monomials is a polynomial.

For example,

$5 a b^{2} -7 a b^{2}$ = $(5-7) a b^{2}$ = $-2 a b^{2}$
$x y + 6 a^{2} b$ (Here, the difference of two monomials is a binomial.)

### Examples on Subtracting Monomials

Given below are some examples that explain how to subtract monomials

Example 1:

Subtracting monomials: $16 a^{2} b^{3} c - 7 a^{2} b^{3} c$

Solution:

Given $16 a^{2} b^{3} c - 7 a^{2} b^{3} c$

Step 1: We subtract the like terms first $(16 - 7)$ and variables $a^{2} b^{3} c$ remain the same.
$(16 - 7) a^{2} b^{3} c$
Step 2: After subtraction we get $9 a^{2} b^{3} c$
Answer: $9 a^{2} b^{3} c$

Example 2:

Subtracting monomials: $28 q - 12 q$

Solution:

Given $28 q - 12 q$

Step 1: We subtract like terms first $(28 - 12)$ and the variable q remains same.
$(28 - 12) q$
Step 2: After subtraction we get $16 q$
Answer: $16 q$

## Multiplying Monomials

There are two different methods of multiplying monomials:

• Multiplication of a monomial by a monomial
• Multiplication of a multinomial by a monomial

### Multiplication of a Monomial by a Monomial

While multiplying a monomial by a monomial, we have to multiply the monomials along with their signs and then simplify them.

### Multiplication of a Multinomial by a Monomial

While multiplying a multinomial by a monomial, we have to multiply each term of the given multinomial by the given monomial along with their signs and then simplify them.

### Examples on Multiplying Monomials

Given below are some examples that explain how to multiply monomials.

Example 1:

Multiply $4a + 3b - 5$ by $2a$

Solution:

$2a \times (4a + 3b - 5)$ = $(2a \times 4a) + (2a \times 3b) - (2a \times 5)$
= $(2 \times 4) (a \times a) + (2 \times 3) (a \times b) - (2 \times 5) a$
= $8a^{2} + 6ab - 10a$
To multiply two monomials, multiply the numerical coefficients separately and the literal coefficients separately with their respective signs

Example 2:

Simplify $mn (m^{2} - n^{2})$

Solution:

$mn (m^{2} - n^{2})$ = $(mn \times m^{2}) - (mn \times n^{2})$
= $(m \times m^{2} \times n) - (n \times n^{2} \times m)$
= $m^{3}n - n^{3}m$

## Dividing Monomials

There are a few steps to be followed in dividing monomials:

Step1: Arrange the monomials in its fractional form, keeping the dividend as a numerator and the divisor as a denominator.
Step2: Divide numerical coefficients.
Step3: Divide literal coefficients.
Step4: Divide the signs.
• (+) ÷ (+) = +
• (+) ÷ (-) = -
• (-) ÷ (-) = +
• (-) ÷ (+) = -
Step5: Multiply the results.

### Examples on Dividing Monomials

Given below are some examples that explain how to divide monomials.

Example 1:

Divide $49a^{3}b^{2}$ by $-7a^{2}b$

Solution:

$49a^{3}b^{2}$ ÷ $-7a^{2}b$ = $\frac{49a^{3}b^{2}}{-7a^{2}b}$ (Arranging the monomials in a fractional form)

= $\frac{-7a^{3}b^{2}}{a^{2}b}$ (Dividing numerical coefficients and signs)
= $-7ab$ (Dividing literal coefficients)

Example 2:

Divide $-32p^{2}qr^{3}$ by $-4pqr$

Solution:

$-32p^{2}qr^{3}$ ÷ $-4pqr$ = $\frac{-32p^{2}qr^{3}}{-4pqr}$ (Arranging the monomials in fractional form)

= $\frac{8p^{2}qr^{3}}{pqr}$ (Dividing numerical coefficients and signs)
= $8pr^{2}$ (Dividing literal coefficients)