Question 1
Solution
Find HCF and LCM of 6x³y² and 21x²y³
First Expression
6x³y²
= 2 × 3 × x × x × x × y × y
Second Expression
21x²y³
= 3 × 7 × x × x × y × y × y
HCF = Common Factors
Common Factors = 3 × x × x × y × y
∴ HCF = 3x²y²
LCM = HCF × Remaining Factors
LCM = 3x²y² × 2 × x × 7 × y
∴ LCM = 42x³y³
Question 2
Solution
Find HCF and LCM of a + 3b and a² – 9b²
First Expression
a + 3b
= (a + 3b)
Second Expression
a² – 9b²
= (a)² – (3b)²
= (a + 3b) × (a – 3b)
HCF = Common Factors
Common Factors = (a + 3b)
∴ HCF = a + 3b
LCM = HCF × Remaining Factors
LCM = (a + 3b) × (a – 3b)
∴ LCM = (a + 3b)(a – 3b) = a² – 9b²
Question 3
Solution
Find HCF and LCM of x² + 6x + 9 and x² + 5x + 6
First Expression
x² + 6x + 9
= x² + 3x + 3x + 9 = x(x + 3) + 3(x + 3)
= (x + 3) × (x + 3)
Second Expression
x² + 5x + 6
= x² + 3x + 2x + 6 = x(x + 3) + 2(x + 3)
= (x + 3) × (x + 2)
HCF = Common Factors
Common Factors = (x + 3)
∴ HCF = x + 3
LCM = HCF × Remaining Factors
LCM = (x + 3) × (x + 3) × (x + 2)
∴ LCM = (x + 3)²(x + 2)
Question 4
Solution
Find HCF and LCM of x² + 2x + 1 and x² – 5x – 6
First Expression
x² + 2x + 1
= x² + x + x + 1 = x(x + 1) + 1(x + 1)
= (x + 1) × (x + 1)
Second Expression
x² – 5x – 6
= x² – 6x + x – 6 = x(x – 6) + 1(x – 6)
= (x + 1) × (x – 6)
HCF = Common Factors
Common Factors = (x + 1)
∴ HCF = x + 1
LCM = HCF × Remaining Factors
LCM = (x + 1) × (x + 1) × (x – 6)
∴ LCM = (x + 1)²(x – 6)
Question 5
Solution
Find HCF and LCM of 25x² + 20x + 4 and 3x² – 2x – 5
First Expression
25x² + 20x + 4
= 25x² + 10x + 10x + 4 = 5x(5x + 2) + 2(5x + 2)
= 1 × (5x + 2) × (5x + 2)
Second Expression
3x² – 2x – 5
= 3x² – 5x + 3x – 5 = x(3x – 5) + 1(3x – 5)
= 1 × (3x – 5) × (x + 1)
HCF = Common Factors
Common Factors = 1
∴ HCF = 1
LCM = HCF × Remaining Factors
LCM = 1 × (5x + 2)² × (3x – 5) × (x + 1)
∴ LCM = (5x + 2)²(3x – 5)(x + 1)
Question 6
IMPORTANT QUESTION
Find HCF & LCM of a² + 2ab + b² – c² and a² – b² – 2bc – c²
First Expression
a² + 2ab + b² – c²
= (a + b)² – c²
= [(a + b) + c][(a + b) – c]
= (a + b + c) × (a + b – c)
Second Expression
a² – b² – 2bc – c²
= a² – (b² + 2bc + c²) = a² – (b + c)²
= [a + (b + c)][a – (b + c)]
= (a + b + c) × (a – b – c)
HCF = Common Factors
Common Factors = (a + b + c)
∴ HCF = a + b + c
LCM = HCF × Remaining Factors
LCM = (a + b + c) × (a + b – c) × (a – b – c)
∴ LCM = (a + b + c)(a + b – c)(a – b – c)