HCF and LCM of Polynomials (original) (raw)

Last Updated : 23 Jul, 2025

HCF (Highest Common Factor) and LCM (Least Common Multiple) of polynomials are concepts similar to those for integers. The HCF of two polynomials is the largest polynomial that divides both polynomials without leaving a remainder, while the LCM is the smallest polynomial that is a multiple of both polynomials.

To find the HCF of polynomials, we take the common factors among all the factors of two polynomials, and for LCM, we take the product of all their unique factors. In this article, we will discuss how to find HCF and LCM for polynomials, with some solved examples as well.

HCF-and-LCM-of-Polynomials

Polynomials

Polynomials are referred to as expressions with multiple terms, including variables, constants, coefficients and exponents. These terms are combined with the addition, subtraction, multiplication or division symbols to give the polynomials. Some examples of polynomial include (x + 2), (p3 + 3p + 9) etc.

What is HCF

**HCF (Highest Common Factor) is the largest number among all the common factors of two or more numbers. Suppose the largest number that divides both x and y is the HCF (Highest Common Factor) of two natural integers, x and y.

**Examples of HCF

Let's take two numbers 24 and 36.

Therefore, the largest number in both lists of factors of 24 and 36 is 12.

How to Find HCF of Polynomials?

Below are the steps to find the HCF of polynomials.

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

**Solution:

First, we do factorization of the given polynomials.

x2 - y2 = (x + y) (x - y)

(x + y)2 = (x + y) (x + y)

Common factor = (x + y)

HCF of the given polynomials = (x + y)

What is LCM?

The smallest number that can be divided by all of the numbers is known as the LCM of two or various numbers. Suppose the smallest that multiply by both 4 and 6 is the LCM (Least common multiple) of two natural integers, 4 and 6.

**Example of LCM

Let's take two numbers again 24 and 36 for finding out the LCM.

Therefore, the smallest number in both lists of multiples of 24 and 36 is 72.

How to Find LCM of Polynomials?

Below are the steps to find the LCM of polynomials.

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

**Solution:

First, we do factorization of the given polynomials.

x2 - y2 = (x + y) (x - y)

(x + y)2 = (x + y) (x + y)

Product of all factors = (x + y)3(x - y)

HCF of the given polynomials = (x + y)3(x - y)

How to Find HCF and LCM of Polynomials

Prime factorization is the most common method to determine the HCF and LCM of algebraic expressions. The following are the steps to take in order to determine the HCF and LCM of an algebraic expression are given below.

**Example: Find the HCF and LCM of the algebraic expressions 6x 3 y 2 , 10x 4 y 4 **and 30x 4 y 3 .

**Solution:

HCF = 2 × x3 × y2 = 2x3 y2

LCM = 30x4 y4

Relationship Between LCM and HCF of Two Polynomials

The product of polynomials equals the product of its H.C.F. and L.C.M., which is the common relationship between L.C.M. and H.C.F. of polynomials. The following is one way to express this relationship.

If p(x) and q(x) are two polynomials, then

**p(x) ∙ q(x) = {H.C.F. of p(x) and q(x)} x {L.C.M. of p(x) and q(x)}

**Example: Find the HCF of p(x) = 3xy and q(x) = 2x 2 **if LCM of p(x) and q(x) is 6x 2 y.

**Solution:

As we know, HCF (_a, _b) × LCM (_a, _b) = _a × _b

⇒ HCF × 6x2y = 3xy × 2x2

⇒ HCF = 3xy × 2x2 / 6x2y

⇒ HCF = x

Thus, HCF of p(x) and q(x) is x.

HCF and LCM Tricks

The HCF and LCM tips can make it simple for students to determine the HCF and LCM of algebraic expressions:

**Read More

Solved Examples of HCF and LCM of Two Polynomials

**Example 1: Find the H.C.F. and L.C.M. of the expressions x 2 – 5x + 6 and x 2 – 7x + 10 by factorization.

**Solution:

x2 - 5x + 6 = x2 - 2x - 3x +6
⇒ x2 - 5x + 6 = 2(x-2) -3(x-2)
⇒ x2 - 5x + 6 = (x - 2) (x - 3)

x2 - 7x + 10 = x2 - 2x -5x + 10
⇒ x2 - 7x + 10 = x(x-2) -5(x-2)
⇒ x2 - 7x + 10 = (x-2) (x-5)

LCM = (x-2) × (x-3) × (x-5)

HCF = (x-2)

**Example 2: Find LCM and HCF of polynomials (x+3) (6x 2 + 5x -4) and (2x 2 **+ 7x + 3) (x + 3).

**Solution:

(x+3) (6x2 + 5x -4) = (x+3) (6x2 + 8x -3x -4)
⇒ (x+3) (6x2 + 5x -4) = (x+3) [2x(3x+4) -1 (3x+4)]
⇒ (x+3) (6x2 + 5x -4) = (x+3) (2x -1) (3x+4)

(2x2 + 7x +3) (x+3) = (2x2 + 6x + x+3) (x+3)
⇒ (2x2 + 7x +3) (x+3) = [2x(x+3) +1 (x+3)] (x+3)
⇒ (2x2 + 7x +3) (x+3) = (2x+1) (x+3) (x+3)
⇒ (2x2 + 7x +3) (x+3) = (2x+1) (x+3)2

LCM = (x+3)2 × (2x-1) ×(3x+4) ×(2x+1)

HCF = (x+3)

**Example 3: Find HCF and LCM of (x 2 + xy + y 2 ) and (x 3 **- y 3 )

**Solution:

HCF = x2 + xy + y2

LCM = (x2 + xy + y2) (x-y) = x3 - y3

**Example 4: Find HCF and LCM of x 2 -9 and x 2 **- 6x + 9.

**Solution:

x2-9 = (x2) - (3)2 = (x-3) (x+3)

x2- 6x + 9 = x2 - 3x -3x +9
⇒ x2- 6x + 9 = x(x-3) -3(x-3)
⇒ x2- 6x + 9 = (x-3) (x-3) or (x-3)2

HCF = (x-3)

LCM = (x-3)2 (x+3)

**Example 5: Determine the HCF and LCM of the polynomials 4a 2 b, 6ab and 8ab 2 .

**Solution:

HCF = 2 × a × b = 2ab

LCM = 23 × 3 × a2 × b2 = 24a2 b2

Practice Questions on HCF and LCM of Two Polynomials

**Q1: Find the HCF and LCM of the expressions 3x2 - 6x+3 and 6x2 -12x +6.

**Q2: Determine the HCF and LCM of the polynomials x4 - 16 and x4 - 4x2 + 4.

**Q3: Find the HCF and LCM of the expressions 9x2 - 16 and 3x2 - 4.

**Q4: Find the HCF and LCM of the polynomials 3x3+ 6x2 - 9x.

**Q5: Determine the HCF and LCM of the polynomials 4x3 - 8x2 and 6x3 - 12x2.