How to Find HCF and LCM of Two Polynomials Step by Step
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.
FAQs on HCF and LCM of Two Polynomials with Methods and Examples
1. What is the HCF and LCM of two polynomials?
The HCF (Highest Common Factor) of two polynomials is the greatest polynomial that divides both exactly, while the LCM (Least Common Multiple) is the smallest polynomial that is exactly divisible by both. In other words:
- HCF = common factors with the smallest powers.
- LCM = all factors (common and uncommon) with the highest powers.
These concepts are similar to HCF and LCM of numbers but applied to algebraic expressions.
2. How do you find the HCF of two polynomials?
To find the HCF of two polynomials, factor each polynomial completely and take the common factors with the lowest powers. Follow these steps:
- Factor each polynomial fully.
- Identify common factors.
- Multiply the common factors with the smallest exponent.
Example: For x² − 9 and x² − 3x,
- x² − 9 = (x − 3)(x + 3)
- x² − 3x = x(x − 3)
Common factor = (x − 3). So, HCF = x − 3.
3. How do you calculate the LCM of two polynomials?
The LCM of two polynomials is found by multiplying all distinct factors with their highest powers after complete factorization. Steps:
- Factor each polynomial completely.
- List all unique factors.
- Choose the highest power of each factor.
- Multiply them together.
Example: For (x − 3)(x + 3) and x(x − 3), the LCM is x(x − 3)(x + 3).
4. What is the formula relating HCF and LCM of two polynomials?
For two polynomials, Product of Polynomials = HCF × LCM (when polynomials are taken in factored form). That is,
- If A(x) and B(x) are polynomials, then
- A(x) × B(x) = HCF × LCM
This relationship is similar to numbers and helps verify answers in algebra problems.
5. Can you give an example of HCF and LCM of two polynomials?
Yes, consider the polynomials x² + 5x + 6 and x² + 3x + 2.
- x² + 5x + 6 = (x + 2)(x + 3)
- x² + 3x + 2 = (x + 1)(x + 2)
Common factor = (x + 2), so HCF = x + 2.
LCM includes all factors: (x + 1)(x + 2)(x + 3), so LCM = (x + 1)(x + 2)(x + 3).
6. What is the difference between HCF and LCM of polynomials?
The HCF is the greatest common divisor of two polynomials, while the LCM is the smallest common multiple. The key differences are:
- HCF: Takes only common factors with smallest powers.
- LCM: Takes all factors with highest powers.
- HCF divides both polynomials exactly.
- LCM is divisible by both polynomials.
7. How do you find the HCF of polynomials using long division?
The HCF of polynomials using long division is found by applying the Euclidean algorithm. Steps:
- Divide the larger polynomial by the smaller one.
- Replace the dividend with the divisor and the divisor with the remainder.
- Repeat until the remainder is zero.
- The last non-zero remainder is the HCF.
This method is useful when factorization is difficult.
8. Is the HCF of two polynomials always unique?
Yes, the HCF of two polynomials is unique up to a constant multiple. In standard form, we usually take the HCF with a positive leading coefficient to make it unique. This ensures a single greatest common divisor for given polynomials.
9. What are common mistakes when finding HCF and LCM of polynomials?
Common mistakes when calculating HCF and LCM of polynomials include:
- Not factoring polynomials completely.
- Ignoring numerical coefficients.
- Choosing highest powers for HCF instead of lowest.
- Forgetting to include all distinct factors in LCM.
Careful factorization and checking using the formula A(x) × B(x) = HCF × LCM helps avoid errors.
10. Why do we need HCF and LCM of two polynomials?
The HCF and LCM of two polynomials are used to simplify algebraic fractions, solve rational expressions, and perform polynomial division. Applications include:
- Simplifying rational algebraic expressions.
- Finding common denominators.
- Solving algebraic equations.
- Working in higher algebra and calculus.
They are fundamental tools in algebra and polynomial operations.
