# Factoring Monomial

In this chapter, we will learn to factorize monomial with solved examples.

In the end we will also learn to find factors of given monomial.

Let us first understand the basics of monomial.

## What is Monomial ?

The algebraic expression with only one term is called monomial.

Note that in algebraic expressions, the terms are separated by addition/subtraction sign.

Hence, the monomial doesn’t contain any terms separated by addition or subtraction.

Examples of Monomials;

\mathtt{( i) \ 5x^{3}}\\\ \\ \mathtt{( ii) \ 10x^{2} y^{4}}\\\ \\ \mathtt{( iii) \ 11\frac{x}{y}}

Given below are examples that are non-monomials.

\mathtt{( i) \ 2+x}\\\ \\ \mathtt{( ii) \ x^{2} -y^{2}}\\\ \\ \mathtt{( iii) \ x^{3} +y^{3} +2xy}

Since all the expression contains more than one terms, they are not monomials.

## How to factorize monomial ?

The breaking of monomial into smaller components is called factoring.

Given below are steps to factorize monomial;

(a) First Factorize coefficient into smaller components.

You can take the help of HCF to factorize the number.

(b) Now factorize the individual variable.

This can be done by converting the exponent of variable into simple multiplication.

The process is easy and straight forward. Let us understand the process with the help of examples.

Example 01
Factorize the monomial \mathtt{6x^{4}}

Solution
First factorize the coefficient of given monomial.

6 ⟹ 2 x 3

Now factorize the variable x.

\mathtt{x^{4} \Longrightarrow \ x.x.x.x}

Joining both coefficient and variable we get;

\mathtt{6x^{4} \Longrightarrow \ 2.3.\ x.x.x.x}

Hence on factorization, we get the individual components of given monomial.

Example 02
Factorize \mathtt{8x^{2} y^{3}}

Solution
First factorize the coefficient.

8 = 2 x 2 x 2

Now factorize the given variables.

Variable x
\mathtt{x^{2} =\ x.\ x}

Variable y
\mathtt{y^{3} =\ y.\ y.\ y}

Joining all the components we get;

\mathtt{8x^{2} y^{3}} = 2.2.2.x.x.y.y.y

Example 03
Factorize \mathtt{30xy^{2}}

Solution

Factorize the coefficient.
30 = 2 x 3 x 5

Now factorize the variables

Variable x
\mathtt{x^{1} \ \Longrightarrow \ x}

Variable y
\mathtt{y^{2} =\ y.\ y}

Joining all the components, we get;

\mathtt{30xy^{2} \Longrightarrow \ 2.3.5.x.y.y}

## Factors of Monomials

The expressions which can completely divide the monomial without leaving any remainder are the factors of monomial.

By factorizing given monomial, we can easily found components which will completely divide the monomial.

Let us understand this with example;

Example 01
Find factors of monomial \mathtt{3x^{2}}

Solution
Do the factorization of given monomial.

\mathtt{3x^{2} \ \Longrightarrow \ 3.x.x}

On grouping different components of monomial, you can get the factors.

Hence, the factors of given monomial are \mathtt{1,\ x,\ x^{2} ,\ 3x\ \&\ 3x^{2}} .

It means that when we divide the monomial with these expression, we would get 0 remainder.

Example 02
Find all the possible factors of monomial \mathtt{10x^{2} y}

Solution
Factorize the given monomial.

\mathtt{10x^{2} y \Longrightarrow \ 2.5.x.x.y}

Now group the different components to get the factor.

The factors are 1, x, \mathtt{x^{2}} , xy, x^{2}y , 2, 2x, \mathtt{\ 2x^{2}} , 2xy, \mathtt{2x^{2} y} , 5, 5x, \mathtt{\ 5x^{2}} , 5xy, \mathtt{5x^{2} y} , 10, 10x, \mathtt{\ 10x^{2}} , 10xy and \mathtt{10x^{2} y}

Example 03
Find factors of monomial 7xy.

Solution
First factorize the monomial.

The monomial cannot be broken down further. Hence, it is already present in factorized form.

The factors of given monomial are x, y, xy, 7x, 7y and 7xy

Next Chapter : Factorization of polynomial by grouping method