HCF and LCM of Polynomials

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.

What are 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.

Example of HCF

Let’s take two numbers 24 and 36.

• Factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
• Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 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.

• First, do factorization of the polynomials (using prime factorization).
• Then, find the common factors or variables.
• The resultant common term gives the HCF of the given polynomials.

Example: Given three polynomials x² – y² 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.

• Multiples of 24 are 24, 48, 72, 96
• Multiples of 36 are 36, 72, 108, 144

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.

• First, do factorization of the polynomials (using prime factorization).
• Then, find the product of all factors or variables.
• The resultant common term gives the LCM of the given polynomials.

Example: Given three polynomials x² – y² 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.

• Firstly, Write down the given algebraic expressions.
• Then, determine which terms belong in each algebraic expression.
• After that, Find the prime factors of each term.
• At last, Take the common of all the prime factors of two polynomials to get the HCF of an algebraic expression and take the product of all of the algebraic expression’s prime components to find the LCM.

Example: Find the HCF and LCM of the algebraic expressions 6x³y², 10x⁴y⁴ and 30x⁴y³.

Solution:

• 6x³ y² = 6 × x³ × y²
• 10x⁴y⁴ = 10 × x⁴ × y⁴
• 30x⁴y³ = 30 × x⁴ × y³

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) = 2x² if LCM of p(x) and q(x) is 6x²y.

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:

• The values of the highest integer in the equation are the HCF and LCM if the expression is a perfect square.
• The LCM is the value of the greatest number less the value of the lowest number in an equation that is not a perfect square.
• The HCF is equal to the sum of the values of the highest and lowest numbers in the expression if it is not a perfect square.

Conclusion: HCF and LCM of Two Polynomials

In conclusion, the highest common factor (HCF) and lowest common multiple (LCM) of any polynomial can be easily calculated by factorizing each polynomial, which enables us to identify the common factors and multiples, providing insights into the fundamental properties and relationships within the polynomials.

Read More,

• HCF and LCM
• Polynomials
• Euclid’s Division Lemma

Solved Examples of HCF and LCM of Two Polynomials

Example 1: Find the H.C.F. and L.C.M. of the expressions x² – 5x + 6 and x² – 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) (6x² + 5x -4) and (2x² + 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 (x² + xy + y²) and (x³ – y³ )

Solution:

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

HCF = x² + xy + y²

LCM = (x² + xy + y²) (x-y) = x³ – y³

Example 4: Find HCF and LCM of x²-9 and x² – 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 4a² b, 6ab and 8ab².

Solution:

• 4a² b = 2²
• 6ab = 2 x 3
• 8ab² = 2³

HCF = 2 × a × b = 2ab

LCM = 2³ × 3 × a² × b² = 24a² b²

HCF and LCM of Two Polynomials: Practice Questions

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.

FAQs on HCF and LCM of Two Polynomials

What is the HCF of two polynomials?

The higher common factor (HCF) of two polynomials is the common polynomial with the lowest degrees and shared coefficients between them.

Can the HCF of two polynomials be zero?

No, the Highest Common Factor (HCF) of two polynomials cannot be zero.

What is the LCM of a Polynomial?

LCM of two polynomials is the product of unique factors with the highest powers from each polynomial.

What is the Relationship Between LCM and HCF?

Yes, It is true that the product of two polynomials’ HCF and LCM equals the polynomials’ product.

HCF × LCM = (1^st Polynomial × 2^nd Polynomial)

Find the HCF of 11ab and 55a.

The HCF of 11ab and 55a is 11a.

What is the LCM of 27pqr and 3p²q²?

The LCM of 27pqr and 3p²q² is 27p² q² r.