# Factorize polynomial with common monomial factor

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 terms.

(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 lowest 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.