# LCM of polynomials

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.