Class 8 RD Sharma Solutions – Chapter 7 Factorization – Exercise 7.5 | Set 1

Factorize each of the following expressions:

Question 1. 16x² – 25y²

Solution:

16x² – 25y²

= (4x)² – (5y)²

= (4x – 5y)(4x + 5y)  

Question 2. 27x² – 12y²

Solution:

27x² – 12y²

= 3(9x² – 4y²)

= 3[(3x)² – (2y)²]

= 3(3x – 2y)(3x + 2y) 

Question 3. 144a² – 289b²

Solution:

144a² – 289b²

= (12a)² – (17b)²

= (12a – 17b)(12a + 17b) 

Question 4. 12m² – 27

Solution:

12m² – 27

= 3(4m² – 9) 

= 3[(2m)² – 3²] 

= 3(2m – 3)(2m + 3) 

Question 5. 125x² – 45y²

Solution:

125x² – 45y²

= 5(25x² – 9y²) 

= 5[(5x)² – (3y)²] 

= 5(5x – 3y)(5x + 3y) 

Question 6. 144a² – 169b²

Solution:

144a² – 169b²

= (12a)² – (13b)²

= (12a – 13b)(12a + 13b)

Question 7. (2a – b)² – 16c²

Solution:

(2a – b)² – 16c²

= (2a – b)² – (4c)²

= [(2a – b) + 4c][(2a – b) – 4c] 

= (2a – b + 4c)(2a – b – 4c) 

Question 8. (x + 2y)² – 4(2x – y)²

Solution:

(x + 2y)² – 4(2x – y)²

= (x + 2y)² – [2(2x – y)]²

= [(x + 2y) – 2(2x – y)][(x + 2y) + 2(2x – y)] 

= (x + 2y – 4x + 2y)(x + 2y + 4x – 2y) 

= 5x(4y – 3x) 

Question 9. 3a⁵ – 48a³

Solution:

3a⁵ – 48a³

= 3a³(a² – 16) 

= 3a³(a² – (4)²) 

= 3a³(a – 4)(a + 4) 

Question 10. a⁴ – 16b⁴

Solution:

a⁴ – 16b⁴

= a⁴ – 2⁴b⁴

= (a²)² – (2²b²)²

= (a² – 2²b²)(a² + 2²b²) 

= [a² – (2b)²](a² + 4b²) 

= [a² – (2b)²](a² + 4b²) 

= (a – 2b)(a + 2b)(a² + 4b²) 

Question 11. x⁸ – 1

Solution:

x⁸ – 1

= (x⁴)² – 1²

= [(x²)² – 1²](x⁴ + 1)

= [(x² – 1)(x² + 1)](x⁴ + 1)

= (x – 1)(x + 1)(x² + 1)(x⁴ + 1) 

Question 12. 64 – (a + 1)²

Solution:

64 – (a + 1)²

= (8)² – (a + 1)²

= [8 – (a + 1)][8 + (a + 1)]  

= (8 – a – 1)(8 + a + 1)  

= (7-a)(9+a)  

Question 13. 36L² – (m + n)²

Solution:

36L² – (m + n)²

= (6L)² – (m + n)²

= [6L – (m + n)][6L + (m + n)]  

= (6L – m – n)(6L + m + n)  

Question 14. 25x⁴y⁴ – 1

Solution:

25x⁴y⁴ – 1

= (5x²y²)² – 1²

= (5x²y² – 1)(5x²y² + 1)  

Question 15. a⁴ – 1/b⁴

Solution:

a⁴ – 1/b⁴

= (a²)² – (1/b²)²

= (a² – 1/b²)(a² + 1/b²)  

= (a – 1/b)(a + 1/b)(a² + 1/b²)  

Question 16. x³ – 144x

Solution:

x³ – 144x

= x(x² – 144)  

= x(x² – 12²)  

= x(x – 12)(x + 12)  

Question 17. (x – 4y)² – 625

Solution:

(x – 4y)² – 625  

= (x – 4y)² – 25²

= [(x – 4y) – 25] [(x – 4y) + 25]  

= (x – 4y – 25)(x – 4y + 25)

Question 18. 9(a – b)² – 100(x – y)²

Solution:

9(a – b)² – 100(x – y)²

= [3(a – b)]² – [10 (x – y)]²

= [3(a – b) -10(x – y)] [3(a – b) + 10(x – y)]

= (3a – 3b – 10x + 10y) (3a – 3b + 10x – 10y)

Question 19. (3 + 2a)² – 25a²

Solution:

(3 + 2a)² – 25a²

= (3 + 2a)² – (5a)²

= (3 + 2a + 5a)(3 + 2a – 5a)

= (3 + 7a)(3 – 3a)

Question 20. (x + y)² – (a – b)²

Solution:

(x + y)² – (a – b)²

= [(x + y) + (a – b)] [(x + y) – (a – b)]

= (x + y + a – b) (x + y – a + b)

Question 21. 1/16 x²y² – 4/49y²z²

Solution:

1/16 x²y² – 4/49y²z²

= (1/4xy)² – (2/7 yz)²

= (1/4xy – 2/7yz) (1/4xy + 2/7yz)

= [y (1/4x – 2/7z)] [y(1/4x + 2/7z)]

= y² (1/4x – 2/7z) (1/4x + 2/7z)

Question 22. 75a³b² – 108ab⁴

Solution:

75a³b² – 108ab⁴

= 3ab²(25a² – 36b²)

= 3ab²[(5a)² – (6b)²]

= 3ab²[(5a – 6b)(5a + 6b)]

= 3ab²(5a – 6b)(5a + 6b)

Question 23. x⁵ – 16x³

Solution:

x⁵ – 16x³

= x³(x² – 16)

= x³(x² – 4²)

= x³(x – 4)(x + 4)

Chapter 7 Factorization – Exercise 7.5 | Set 2