In this chapter, we will learn to find LCM of polynomials with solved examples.

To understand the chapter, you should have basic knowledge of concept of LCM.

## How to find LCM of polynomials ?

The LCM of polynomial is the **lowest multiple which gets fully divided by given polynomials**.

To find the LCM, **follow the below steps**;

(a) **Factorize the given polynomials** into smaller factors.

(b) **Compare the common factors** and **select the one with highest power**.

(c) Now **select the uncommon factors**.

(d) Now** combine the common and uncommon factors** and you will get the LCM

I hope you understand the above process. Let us see some solved examples for further clarity.

## LCM of polynomials – Solved Problems

**Example 01**

Find the LCM of below polynomials.

\mathtt{\Longrightarrow \ \left( x^{2} +4x+4\right) \ ( x+3) \ \ \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \left( x^{2} +2x\right)}

**Solution**

For LCM, follow the below steps;

**(a) Factorize each of the polynomial into small factors.**

(i) \mathtt{\left( x^{2} +4x+4\right) \ ( x+3) \ } \\\ \\

\mathtt{\Longrightarrow \ ( x+2)^{2}( x+3)}

(ii) \mathtt{\left( x^{2} +2x\right)} \\\ \\

\mathtt{\Longrightarrow x\ ( x\ +\ 2)}

**(b) Find common and uncommon factors with highest power.**

Given below is the factorized form of above polynomials.

(i) \mathtt{ \ ( x+2)^{2}( x+3)}\\\ \\

(ii) \mathtt{ x\ ( x\ +\ 2)}

**Common factor is (x + 2)**

The common factor with highest power is \mathtt{( x+2)^{2}}

**Non common factors are x . (x + 3)**

**Combining common and non common factors, we get;**

\mathtt{\Longrightarrow \ x.\ ( x+3) .( x+2)^{2}}\\\ \\ \mathtt{\Longrightarrow \ x( x+3)( x+2)^{2}}

Hence, the above expression is the LCM of given polynomials.

**Example 02**

Find the LCM of given polynomials.

\mathtt{\Longrightarrow \ x^{3} +3x^{2} -4x-12}\\\ \\ \mathtt{\Longrightarrow \ \left( x^{2} +6x+9\right)}

**Solution****First factorize each polynomial into smaller factors.**

( i ) \mathtt{x^{3} +3x^{2} -4x-12}

\mathtt{\Longrightarrow \ x^{2}( x+3) -4( x+3)}\\\ \\ \mathtt{\Longrightarrow \ \left( x^{2} -4\right)( x+3)}\\\ \\ \mathtt{\Longrightarrow \ ( x-2)( x+2)( x+3)}

\mathtt{( ii) \ \left( x^{2} +6x+9\right)}\\\ \\

\mathtt{\Longrightarrow \ x^{2} +2.3.x+3^{2}}

Referring the formula;

\mathtt{( a+b)^{2} =a^{2} +2ab\ +\ b^{2}}

Using the formula we get;

\mathtt{\Longrightarrow \ ( x\ +\ 3)^{2}}

(b) **Find common and uncommon factors with highest power**.

Writing the factorized form of given polynomials.

\mathtt{\Longrightarrow \ ( x-2)( x+2)( x+3)}\\\ \\ \mathtt{\Longrightarrow \ ( x\ +\ 3)^{2}}

**Common factor is ( x + 3)**

**Common factor with highest power** is \mathtt{( x\ +\ 3)^{2}} .

**Uncommon factors are (x – 2) (x + 2)**

**Combining common and uncommon factors** we get;

\mathtt{\Longrightarrow ( x-2)( x+2) \ ( x\ +\ 3)^{2}}

Hence, the above expression is HCF of given polynomials.

**Example 03**

Find the LCM of given polynomials.

\mathtt{\Longrightarrow \ ( x+2)^{2}( x+5)^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( x+2)^{4}( x+7)}\\\ \\ \mathtt{\Longrightarrow \ ( x+5)^{2}( x+7)^{2}( x+8)}

**Solution**

All the given polynomials are already factorized, so we don’t need to do any further simplification.

Note that there is** no common factor present** in all three polynomials.

Now **we will check individual uncommon factor and select its highest power**.

There are 4 different types of factor present in the given polynomials, (x + 2), (x + 5), (x + 7), (x + 8).

Finding the highest power of each factors.**Highest power of (x + 2)** ⟹ 4**Highest power of (x + 5)** ⟹ 3

**Highest power of (x + 7)** ⟹ 2

**Highest power of (x + 8) **⟹ 1

Combining all the factors with the highest power we get;

\mathtt{\Longrightarrow ( x\ +2)^{4} .( x+5)^{3} .( x+7)^{2} .( x+8)}

Hence, the above expression if the LCM of given polynomials.

**Example 04**

Find the LCM of below polynomials.

\mathtt{\Longrightarrow \ 12\ ( x-3)^{4} \ ( x+4)}\\\ \\ \mathtt{\Longrightarrow \ 10\ ( x+4)^{2}( x+7)^{3}}

**Solution**

The factorized form of both polynomials are expressed as;

\mathtt{\Longrightarrow \ 2^{2} \times 3\times \ ( x-3)^{4} \ ( x+4)}\\\ \\ \mathtt{\Longrightarrow \ 2\times 5\times \ ( x+4)^{2}( x+7)^{3}}

**Let’s first find the LCM of constant terms.**

The constant term of both polynomial is expressed below;

\mathtt{\Longrightarrow \ 2^{2} \times 3}\\\ \\ \mathtt{\Longrightarrow \ 2\times 5}

The LCM is given as;

\mathtt{LCM\ \Longrightarrow \ 2^{2} \times 3\times 5}\\\ \\ \mathtt{LCM\ \Longrightarrow \ 60}

Hence, **the LCM of constant term is 60**.

Now **let’s find the LCM of variables.**

Here (x + 4) is the common factor among the given polynomial.

**Common factor with highest power is** \mathtt{( x+4)^{2}}

**Uncommon factors with highest power are**;

\mathtt{\Longrightarrow \ ( x-3)^{4} \ ( x+7)^{3}}

**Combining the constant value LCM, common and uncommon values we get**;

\mathtt{\Longrightarrow 60\ ( x+4)^{2} \ ( x-3)^{4} \ ( x+7)^{3}}

**Example 05**

Find the LCM of below polynomials

\mathtt{\Longrightarrow \ 6\ ( x+10) \ \left( x^{2} +10x+25\right)}\\\ \\ \mathtt{\Longrightarrow \ 9\ \left( x^{2} +9x+20\right)}

**Solution**

First factorize each of the polynomials.

(i) 6 (x+10) (x^{2}+10x+25)

\mathtt{\Longrightarrow \ 2\times 3\ ( x+10)\left( x+2.x.5+5^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ 2\times 3\ ( x+10) \ ( x+5)^{2}}

(ii) \mathtt{9\ \left( x^{2} +9x+20\right)}

\mathtt{\Longrightarrow \ 3^{2}\left( x^{2} +5x+4x+20\right)}\\\ \\ \mathtt{\Longrightarrow \ 3^{2}( x( x+5) +4( x+5))}\\\ \\ \mathtt{\Longrightarrow \ 3^{2}( x\ +4)( x+5)}

**Given below is the factorized form of both polynomials.**

(i) \mathtt{\ 2\times 3\ ( x+10) \ ( x+5)^{2}} \\\ \\

(ii) \mathtt{\ 3^{2}( x\ +4)( x+5)}

**Let’s first find the LCM of constant terms.**

The constant term of polynomial is given below;

\mathtt{( i) \ \ 2\times 3}\\\ \\ \mathtt{( ii) \ \ 3^{2}}

**Given below is the LCM calculation**

\mathtt{LCM\ =\ 2\times 3^{2}}\\\ \\ \mathtt{LCM\ =\ 18}

**Select common factor with highest power.**

( x + 5) is the common factor.

Highest power of (x + 5) in given polynomials is 2.

Hence, \mathtt{( x+5)^{2}} is the part of LCM.

**Select uncommon factor with highest power**.

Here (x + 4) & (x + 10) are uncommon factors.

Hence, both are part of LCM.

**Combining LCM of constants, Common and uncommon factors we get;**

\mathtt{\Longrightarrow \ 18\ ( x+5)^{2}( x+4)( x+10)}

Hence, given expression is the LCM.