HCF and LCM of Two Polynomials

Do you find working with LCM and HCF exciting? The Least Common Multiple is abbreviated as LCM. It is defined as the smallest number that, when multiplied by both of the given numbers, gives a product that is less than or equal to the original number. The Highest Common Factor is abbreviated as HCF. It is the largest positive number that divides the given numbers. The LCM and HCF of 3, 4, and 5 are 60 and 1, respectively. In this article, you will learn how to find the HCF and LCM of algebraic expressions, the relation between HCF and LCM of two polynomials, and HCF and LCM tricks, along with some solved examples of HCF and LCM.

Finding the HCF and LCM of Algebraic Expressions

The most convenient method to find the HCF and LCM of algebraic expressions is prime factorisation. Steps to be followed while finding the HCF and LCM of algebraic expression are given below:

• Write the given algebraic expressions

• Separate terms corresponding to each algebraic expression

• Find the prime factors of all the terms

• To obtain the HCF of algebraic expression, take the common of all the prime factors of two polynomials.

• To obtain the LCM of algebraic expression, take the product of all its prime factors.

For example, the HCF and LCM of algebraic expressions $2 \mathrm{y}$ and $14 x \mathrm{y}^3$ are given by writing their prime factors, i.e. 2, $y$ and $(7)(2), x,(y)(y)(y)$ respectively.

HCF of the algebraic expressions is $2 \mathrm{y}$

LCM of the algebraic expression is $14 x y^3$.

What is the Relation Between LCM and HCF of Two Polynomials?

The relation between LCM and HCF of two polynomials is given below:

The product of two polynomials is equal to the product of their LCM and $\mathrm{HCF}$, i.e. $\mathrm{p}(\mathrm{x}) \times$ $q(x)=\{$ L.C.M of $p(x)$ and $q(x)\} \times \{$ H.C.F of $p(x)$ and $q(x)\}$, where $p(x)$ and $q(x)$ are two polynomials.

For example: Let the two polynomials be $p(x)=3 x y$ and $q(x)=2 x^2$ LCM of $p(x)$ and $q(x)$ is $6 x^2 y$

HCF of $p(x)$ and $q(x)$ is $x$

The product of two polynomials, $p(x)$ and $q(x)$ is $3 x y \times 2 x^2=6 x^3 y \quad$ ____(1)

Product of their HCF and LCM is given by $x \times 6 x^2 y=6 x^3 y \quad$ ____(2)

Equations 1 and 2 show that the product of two polynomials is equal to their HCF and LCM products.

HCF and LCM of Two Polynomials

In the above image, the number represents the polynomials.

HCF and LCM Tricks

Here are some HCF and LCM tricks that can help students to find the HCF and LCM of algebraic expressions easily:

• If the expression is a perfect square, the HCF and LCM are the value of the highest number in the expression.

• If the expression is not a perfect square, the HCF is the value of the highest number plus the value of the lowest number.

• If the expression is not a perfect square, the LCM is the value of the highest number minus the value of the lowest number.

Solved Examples of HCF and LCM

Q 1. Find the LCM and HCF of 9a³b² and 15b³c²

Ans: Find the LCM and HCF of $9 a^3 b^2$ and $15 b^3 c^2$. Here,

Prime factors of 1st expression, $9 a^3 b^2$ are $3 \times 3 \times a \times a \times a \times b \times b$

Prime factors of 2nd expression, $15 b^3 c^2$ are $3 \times 5 \times b \times b \times b \times c \times c$

HCF $=3 \times \mathrm{b} \times \mathrm{b}=3 \mathrm{~b}^2$

LCM $=3 \times 3 \times 5 \times a \times a \times a \times b \times b \times b \times c \times c=45 a^3 b^3 c^2$

Thus the LCM and HCF of $9 a^3 b^2$ and $15 b^3 c^2$ are $45 a^3 b^3 c^2$ and $3 b^2$, respectively.

Q 2. Find the HCF of 6xy²z, 8x²y³z², 12x³y³z³.

Ans: The prime factors are as following:

$\Rightarrow 6 x y^2 z=2 \times 3 \times x \times y \times y \times z$

$\Rightarrow 8 x^2 y^3 z^2=2 \times 2 \times 2 \times x \times x \times y \times y \times y \times z \times z$

$\Rightarrow 12 x^3 y^3 z^3=2 \times 2 \times 3 \times x \times x \times x \times y \times y \times y \times z \times z \times z$

HCF of $6 x y^2 z, 8 x^2 y^3 z^2 \text { and } 12 x^3 y^3 z^3=2 \times x \times y \times y \times z=2 x y^2 z$

Practice Questions

Q 1. Find the LCM and HCF of 21xy² and 63x²y.

Ans: HCF: 21xy

LCM: $63 x^2 y^2$

Q 2. Find the HCF of 11ab and 55a.

Ans: $11 a$

Q 3. What is the LCM of 27pqr and 3p²q² ?

Ans: $27 p^2 q^2 r$

Summary

This article has covered the most important topic used in every mathematical calculation. Here we have learned how to find the HCF and LCM of algebraic expressions, what is the relation between LCM and HCF, and the HCF and LCM tricks. The relation between HCF and LCM of two polynomials states that the product of two polynomials is equal to that of their LCM and HCF. We also discussed some solved examples of HCF and LCM, along with some practice questions that the students need to solve by themselves for a better understanding of the concept.