HCF & LCM OF BINOMIALS AND TRINOMIALS

Introduction

Just as we find the Highest Common Factor (HCF) and Least Common Multiple (LCM) of numbers, we can find them for algebraic expressions. This requires first factorising the expressions completely.

Brief Overview

1. Finding the HCF of Polynomials

• Definition: The HCF is the product of the common factors of the given polynomials, each raised to its lowest power.
• Method:
  1. Factorise each polynomial completely.
  2. Identify the common factors.
  3. The HCF is the product of these common factors.

Numerical Example:
Find the HCF of $(x^2 - 4)$ and $(x^2 + 4x + 4)$.
Factor 1: $x^2 - 4 = (x - 2)(x + 2)$.
Factor 2: $x^2 + 4x + 4 = (x + 2)^2$.
The only common factor is $(x + 2)$.
HCF = $(x + 2)$.

MCQs - HCF

1. The first step in finding the HCF of two polynomials is to:
   a) Multiply them
   b) Add them
   c) Factorise them completely
   d) Divide them
   Answer: c
2. What is the HCF of $x(x-1)$ and $x^2(x-1)^2$?
   a) $x^2(x-1)^2$
   b) $x(x-1)$
   c) $x$
   d) $x-1$
   Answer: b
3. Find the HCF of $a^2 - b^2$ and $a + b$.
   a) $a - b$
   b) $a + b$
   c) $a^2 - b^2$
   d) 1
   Answer: b (Because $a^2-b^2 = (a+b)(a-b)$)
4. The HCF of $2x + 4$ and $3x + 6$ is:
   a) $x+2$
   b) $6(x+2)$
   c) $x+4$
   d) $2x+6$
   Answer: a (Factors are $2(x+2)$ and $3(x+2)$)
5. If two polynomials have no common factors other than 1, their HCF is:
   a) 0
   b) 1
   c) Their product
   d) Cannot be determined
   Answer: b
6. Find the HCF of $3x^2y$ and $6xy^2$.
   a) $3x^2y^2$
   b) $18x^3y^3$
   c) $3xy$
   d) $xy$
   Answer: c
7. What is the HCF of $x^2 + 5x + 6$ and $x^2 + 4x + 3$?
   a) $x+2$
   b) $x+3$
   c) $x+1$
   d) $(x+3)(x+2)$
   Answer: b (Factors are $(x+2)(x+3)$ and $(x+1)(x+3)$)
8. The HCF of $a^3$ and $a^4$ is:
   a) $a$
   b) $a^3$
   c) $a^4$
   d) $a^7$
   Answer: b
9. Find the HCF of $4x(x-y)^2$ and $6x^2(x-y)$.
   a) $12x^2(x-y)^2$
   b) $2x(x-y)$
   c) $x(x-y)$
   d) $2x^2(x-y)^2$
   Answer: b
10. The HCF of $p^2 - 1$ and $p^3 - 1$ is:
    a) $p-1$
    b) $p+1$
    c) $p^2-1$
    d) $p^2+p+1$
    Answer: a (Factors are $(p-1)(p+1)$ and $(p-1)(p^2+p+1)$)

2. Finding the LCM of Polynomials

• Definition: The LCM is the product of all unique factors of the given polynomials, each raised to its highest power.
• Method:
  1. Factorise each polynomial completely.
  2. List every unique factor that appears in any of the polynomials.
  3. The LCM is the product of these factors, taking the highest power of each.

Numerical Example:
Find the LCM of $(x^2 - 9)$ and $(x^2 - 6x + 9)$.
Factor 1: $x^2 - 9 = (x - 3)(x + 3)$.
Factor 2: $x^2 - 6x + 9 = (x - 3)^2$.
Unique factors: $(x - 3)$ and $(x + 3)$.
Highest power of $(x - 3)$ is 2. Highest power of $(x + 3)$ is 1.
LCM = $(x - 3)^2(x + 3)$.

MCQs - LCM

1. To find the LCM, you take the common and non-common factors raised to their:
   a) Lowest power
   b) Highest power
   c) Sum of powers
   d) Difference of powers
   Answer: b
2. What is the LCM of $x^2$ and $x^3$?
   a) $x$
   b) $x^2$
   c) $x^3$
   d) $x^5$
   Answer: c
3. Find the LCM of $(x-1)$ and $(x+1)$.
   a) 1
   b) $(x-1)$
   c) $(x+1)$
   d) $x^2 - 1$
   Answer: d (The product $(x-1)(x+1)$)
4. The LCM of $2(x+y)$ and $4(x+y)^2$ is:
   a) $2(x+y)$
   b) $4(x+y)$
   c) $8(x+y)^3$
   d) $4(x+y)^2$
   Answer: d
5. The HCF of two polynomials is $(x-2)$ and their LCM is $(x-2)(x+3)$. If one polynomial is $(x-2)(x+3)$, the other is:
   a) $(x+3)$
   b) $(x-2)$
   c) $(x-2)^2$
   d) $(x+3)^2$
   Answer: b
6. What is the LCM of $2ab$ and $3bc$?
   a) $6abc$
   b) $6ab^2c$
   c) $ab$
   d) $b$
   Answer: a
7. Find the LCM of $x^2 - y^2$ and $(x - y)^2$.
   a) $(x-y)$
   b) $x^2-y^2$
   c) $(x-y)(x+y)^2$
   d) $(x+y)(x-y)^2$
   Answer: d
8. The LCM of $x^3$ and $x^5$ is:
   a) $x^3$
   b) $x^5$
   c) $x^8$
   d) $x^2$
   Answer: b
9. What is the LCM of $(a+b)$ and $(a^2 - ab + b^2)$?
   a) $a^3 - b^3$
   b) $a^3 + b^3$
   c) $(a+b)^3$
   d) $a^2 + b^2$
   Answer: b
10. If the product of two expressions is $x^3 - 8$ and their HCF is $x-2$, then their LCM is:
    a) $x+2$
    b) $x^2+2x+4$
    c) $x^2-2x+4$
    d) $x-2$
    Answer: b (Because LCM = Product / HCF = $(x^3 - 8) / (x - 2)$)

Fun Facts about Algebraic HCF & LCM!

• The Golden Rule: Just like with regular numbers, the product of two polynomials is equal to the product of their HCF and LCM! $P(x) \times Q(x) = \text{HCF}(P,Q) \times \text{LCM}(P,Q)$.
• Rational Expressions: HCF is primarily used for reducing algebraic fractions to their lowest terms, while LCM is essential for finding common denominators to add or subtract algebraic fractions.