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.

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

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

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 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.)

Examples on Adding Monomials

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: $23 p$

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: $16 q$

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:

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

    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)