LCM of Polynomials (original) (raw)

Last Updated : 23 Jul, 2025

LCM (Least Common Multiple) of two or more numbers is the smallest number that can be evenly divided by all of the respective numbers. In the case of polynomials, the LCM refers to the smallest polynomial expression that can be evenly divided by the given polynomials.

**Example: Let's take two polynomials : ****(x** 2 + 2x) and (x 2 − 4) .

By doing factorization
(x2 + 2x) = x (x − 2).
(x2 − 4) = (x + 2)(x - 2).

**LCM( (x 2 + 2x) and (x 2 − 4) ) = x(x + 2)(x − 2) = (x3 - 4x)

How to Find LCM of Polynomials?

While finding the LCM of polynomials the goal is to find the smallest polynomial that can be evenly divided by the given polynomials, just as the LCM of numbers is the smallest number divisible by each of the given numbers. This involves factoring the polynomials and selecting the highest powers of their factors to create the LCM.

To find the LCM of polynomials, follow these steps:

• **Step 1: Factor each polynomial into its simplest factors (e.g., linear or quadratic factors).
• **Step 2: Identify all the unique factors from both polynomials.
• **Step 3: For each factor, choose the highest power of that factor from both polynomials.
• **Step 4: Multiply all the selected factors together to get the LCM.

Let's consider an example for better understanding.

**Example: Given three polynomials x 2 - y 2 and (x + y) 2 . Find the HCF of these polynomials.
**Solution:

First, we factorize the given polynomials.

**x 2 - y 2 = (x + y) (x - y)
(x + y)2 = (x + y) (x + y)

Unique factors : (x + y), (x - y)

Product of the highest power all factors = (x + y)2(x - y)1

LCM of the given polynomials = ****(x + y)** 2 (x - y)

**Read More,

• **HCF and LCM
• **HCF of Polynomials
• **Factorization of Polynomials

Solved Example on LCM of Polynomials

**Example 1: Find the LCM of x 3 **− 2x 2 + x and x 2 −x.
**Solution:

Factorize the given polynomials.

**x 3 **− 2x 2 + x = x(x−1)(x−1)
**x 2 −2x = x(x−2)

Unique factors : x, (x − 1), (x - 2)

Product of the highest power all factors = x(x−1)(x−1)(x - 2)

LCM of the given polynomials = **x 4 - 4x 3 **+ 5x 2 -2x

**Example 2: Find the LCM of x 2 **+ 3x + 2 and x 2 **− 1.
**Solution:

Factorize the given polynomials.

**x 2 **+ 3x + 2 = (x + 1)(x + 2)
**x 2 −1 = (x + 1)(x − 2)

Unique factors : (x +1 ), (x + 2), (x - 2)

Product of the highest power all factors = (x + 1)(x + 2)(x - 2)

LCM of the given polynomials = x 3 **+ x 2 - 4x - 4

**Example 2: Find the LCM of x 2 **+ 5x + 6 and x 2 **+ 2x − 3.
**Solution:

Factorize the given polynomials.

x 2 **+ 5x + 6= (x + 2)(x + 3)
**x 2 **+ 2x − 3. = (x - 1)(x + 3)

Unique factors : (x - 1 ), (x + 2), (x + 3)

Product of the highest power all factors = (x - 1)(x + 2)(x + 3)

LCM of the given polynomials = x3 + 4x2 + x - 6