In this chapter we will learn to factorize polynomial containing common monomial factor.

Here I will show you the step by step process to factorize the polynomial along with solved examples at the end.

## Factorizing polynomial with monomial factor

Polynomial is an expression which **contain two or more terms separated by addition or subtraction sign**.

**Monomial is a single expression term**.

If the polynomial has the common monomial in all the terms, we can separate the monomial from the rest of the expressions.**Given below are steps for factorization**;

Let us consider the polynomial for reference.

\mathtt{\Longrightarrow \ 6x^{2} y+12xy+18xy^{2}}

(a) **Take the HCF **of all the constants present.

HCF (6, 12, 18) = 6

Hence, number 6 is the common number present in all the terms of polynomial.

We can easily separate number 6 from the given polynomial.

(b) **Finding common variables**.

For each variable available in polynomial, select the one with lowest exponent value.**Variable x**

\mathtt{\ 6x^{2} y} ⟹ power of x is 2

12xy ⟹ power of x is 1

\mathtt{18xy^{2}} ⟹ power of x is 1

Here the **lowest power of x is 1.**

It means that you can separate x from the the polynomial.

**Variable y**

\mathtt{\ 6x^{2} y} ⟹ power of y is 1

12xy ⟹ power of y is 1

\mathtt{18xy^{2}} ⟹ power of y is 2

**Lowest power of y is 1.**

Hence, you can separate variable y from the polynomial.

Hence, **6xy is the common monomial factor of given polynomial**.**(c) Separating the constant and variables**

**Separating 6xy from the polynomial and divide each term by 6xy**.

\mathtt{\Longrightarrow \ 6x^{2} y+12xy+18xy^{2}}\\\ \\ \mathtt{\Longrightarrow \ 6xy\ \left(\frac{6x^{2} y}{6xy} +\frac{12xy}{6xy} +\frac{18xy^{2}}{6xy}\right)}\\\ \\ \mathtt{\Longrightarrow \ 6xy\ ( \ x+2+3y)}

Hence, the above term is factorized form of polynomial.

I hope you understood the above process. Let us see some examples for further reference.

**Example 01**

Factorize the below expression.

\mathtt{\Longrightarrow \ 25x +75x^{2} +\ 250y^{2}}

**Solution****(a) HCF of all constants**.

HCF (25, 75, 250) = 25

It tells that number 25 is present in all terms.**(b) Lowest power of all variables****Variable x**

25x ⟹ power of x is 1

\mathtt{75x^{2}} ⟹ power of x is 2

\mathtt{250y^{2}} ⟹ power of x is 0

**Lowest power of x is 0.****Variable y**

25x ⟹ power of y is 0

\mathtt{75x^{2}} ⟹ power of y is 0

\mathtt{250y^{2}} ⟹ power of y is 2

**Lowest power of y is 0.**

Hence, the common monomial is \mathtt{25x^{0} y^{0\ } =25}

**(c) Separate the common monomial**

Separate number 25 from polynomial and divide each term by 25.

\mathtt{\Longrightarrow \ 25x +75x^{2} +\ 250y^{2}}\\\ \\ \mathtt{\Longrightarrow \ 25\ \left(\frac{25x}{25} +\frac{75x^{2}}{25} +\frac{250y^{2}}{25}\right)}\\\ \\ \mathtt{\Longrightarrow \ 25\ \left( x+3x^{2} +10y^{2}\right)}

**Example 02**

Factorize the below expression

\mathtt{\Longrightarrow \ 13x^{5} +2x^{3}} **Solution****(a) HCF of all constants**

HCF (13, 2) = 1

**Number 1 is the constant among all the term**s.**(b) Lowest power of all variables****Variable x**

\mathtt{13x^{5}} ⟹ Power of x is 5

\mathtt{2x^{3}} ⟹ Power of x is 3

The l**owest power of x is 3.**

Hence, the common monomial is \mathtt{1.x^{3} =x^{3}}

(c) **Separate the common monomial**

Separate \mathtt{x^{3}} from polynomial and divide each term by \mathtt{x^{3}} .

\mathtt{\Longrightarrow \ 13x^{5} +2x^{3}}\\\ \\ \mathtt{\Longrightarrow \ x^{3}\left(\frac{13x^{5}}{x^{3}} +\frac{2x^{3}}{x^{3}}\right)}\\\ \\ \mathtt{\Longrightarrow \ x^{3} \ \left( 13x^{2} +2\right)}

Hence, the above term is factorized form of given polynomial

**Example 03**

Factorize the polynomial

\mathtt{\Longrightarrow \ 4x^{2} y^{3} z^{3} +6x^{4} y^{5} z^{2} +2x^{3} y^{4} z{^{5}}} **Solution****(a) Find HCF of constant**

HCF (4, 6, 2) = 2

Hence,** number 2 is the common factor of all the terms of polynomial**.**(b) Lowest power of all variables****Variable x**

\mathtt{4x^{2} y^{3} z^{3}} ⟹ Power of x is 2

\mathtt{6x^{4} y^{5} z^{2}} ⟹ Power of x is 4

\mathtt{2x^{3} y^{4} z^{5}} ⟹ Power of x is 3

Here the **lowest power of x is 2.****Variable y**

\mathtt{4x^{2} y^{3} z^{3}} ⟹ Power of y is 3

\mathtt{6x^{4} y^{5} z^{2}} ⟹ Power of y is 5

\mathtt{2x^{3} y^{4} z^{5}} ⟹ Power of y is 4

The **lowest power of y is 3.****Variable z **

\mathtt{4x^{2} y^{3} z^{3}} ⟹ Power of z is 3

\mathtt{6x^{4} y^{5} z^{2}} ⟹ Power of z is 2

\mathtt{2x^{3} y^{4} z^{5}} ⟹ Power of z is 5

**Lowest power of z is 2.**

Hence the monomial \mathtt{2x^{2} y^{3} z^{2}} is the common monomial in the given polynomial.**(c) Separate common monomial**

Separate \mathtt{2x^{2} y^{3} z^{2}} from the monomial and divide each term by same common monomial.

\mathtt{\Longrightarrow \ 4x^{2} y^{3} z^{3} +6x^{4} y^{5} z^{2} +2x^{3} y^{4} z{^{5}}}\\\ \\ \mathtt{\Longrightarrow 2x^{2} y^{3} z^{2}\left(\frac{\mathtt{4x^{2} y^{3} z^{3}}}{\mathtt{2x^{2} y^{3} z^{2}}} +\frac{\mathtt{6x^{4} y^{5} z^{2}}}{\mathtt{2x^{2} y^{3} z^{2}}} +\frac{\mathtt{2x^{3} y^{4} z^{5}}}{\mathtt{2x^{2} y^{3} z^{2}}}\right)}\\\ \\ \mathtt{\Longrightarrow 2x^{2} y^{3} z^{2}\left( \ 2z+3x^{2} y^{2} +xyz^{3}\right) \ }

**Example 04**

\mathtt{\Longrightarrow \ 11x{^{10}} +1331x^{13}}

**Solution****(a) Find HCF of constants**

HCF (11, 1331) = 11

Hence, **number 11 is present in all terms of polynomial.****(b) Lowest power of all variables**

\mathtt{11x{^{10}}} ⟹ power of x is 10

\mathtt{1331x^{13}} ⟹ power of x is 13

**Lowest power of x is 10.**

Hence, the monomial \mathtt{11x^{10}} is common among all the term of the polynomial.**(c) Separating the common terms**

Separate \mathtt{11x^{10}} from the monomial and divide each term by same common monomial.

\mathtt{\Longrightarrow \ 11x{^{10}} +1331x^{13}}\\\ \\ \mathtt{\Longrightarrow \ 11x^{10}\left(\frac{11x^{10}}{11x^{10}} +\frac{1331x^{13}}{11x^{10}}\right)}\\\ \\ \mathtt{\Longrightarrow \ 11x^{10}\left( 1+121x^{3}\right)}

Hence, the above expression is factorized form of polynomial.