Class 8 RD Sharma Solutions – Chapter 7 Factorization-Exercise 7.5 | Set 2

Chapter 7 Factorization – Exercise 7.5 | Set 1

Factorize each of the following expressions:

Question 24. 50/x² – 2x²/81

Solution:

50/x² – 2x²/81

= 2(25/x² – x²/81)

= 2[(5/x)² – (x/9)²]

= 2(5/x – x/9) (5/x + x/9)

Question 25. 256x⁵ – 81x

Solution:

256x⁵ – 81x

= x(256x⁴ – 81)

= x[(16x²)² – (9)²]

= x(16x² – 9) (16x² + 9)

Question 26. a⁴ – (2b + c)⁴

Solution:

a⁴ – (2b + c)⁴

= (a²)² – [(2b + c)²]²

= [a² – (2b + c)²] [a² + (2b + c)²]

= [a + (2b + c)] [a – (2b + c)] [a² + (2b + c)²]

= (a + 2b + c) (a – 2b – c) [a² + (2b + c)²]

Question 27. (3x + 4y)⁴ – x⁴

Solution:

(3x + 4y)⁴ – x⁴

= [(3x + 4y)²]² – (x²)²

= [(3x + 4y)² – x²] [(3x + 4y)² + x²]

= [(3x + 4y) – x] [(3x + 4y) + x] [(3x + 4y)² + x²]

= (2x + 4y) (4x + 4y) [(3x + 4y)² + x²]

= 8(x + 2y) (x + y) [(3x + 4y)² + x²]

Question 28. p²q² – p⁴q⁴

Solution:

p²q² – p⁴q⁴

= p²q² (1 – p²q²)

= p²q² (1 + pq) (1 – pq)

Question 29. 3x³y – 243xy³

Solution:

3x³y – 243xy³

= 3xy (x² – 81y²)

= 3xy [x² – (9y)²]

= 3xy (x – 9y) (x + 9y)

Question 30. a⁴b⁴ – 16c⁴

Solution:

a⁴b⁴ – 16c⁴

= (a²b²)² – (4c²)²

= (a²b² – 4c²) (a²b² + c²)

= [a²b² – (2c)²] (a²b² + c²)

= (ab – 2c) (ab + 2c) (a²b² + c²)

Question 31. x⁴ – 625

Solution:

x⁴ – 625

= (x²)² – (25)²

= (x² – 25) (x² + 25)

= (x² – 5²) (x² + 25)

= (x – 5) (x + 5) (x² + 25)

Question 32. x⁴ – 1

Solution:

x⁴ – 1

= (x²)² – 1²

= (x² – 1)(x² +1)

= (x + 1)(x – 1) (x² + 1)

Question 33. 49(a – b)² – 25(a + b)²

Solution:

49(a – b)² – 25(a + b)²

= [7(a – b)]² – [5 (a + b)]²

= [7(a – b) – 5 (a + b)] [7(a – b) + 5 (a + b)]

= (7a – 7b -5a – 5b) (7a – 7b + 5a + 5b)

= (2a – 12b) (12a – 2b)

= 4(a – 6b)(6a – b)

Question 34. x – y – x² + y²

Solution:

x – y – x² + y²

= (x – y) – (x² – y²)

= (x – y) – (x  – y)(x + y)

= (x – y) [1 – (x + y)]

= (x – y) (1 – x – y)

Question 35. 16(2x – 1)² – 25y²

Solution:

16(2x – 1)² – 25y²

= [4(2x – 1)]² – (5y)²

= [4(2x – 1) – 5y] [4(2x – 1) + 5y]

= (8x – 4 – 5y) ( 8x – 4 + 5y)

Question 36. 4(xy + 1)² – 9(x – 1)²

Solution:

4(xy + 1)² – 9( x- 1)²

= [2(xy + 1)]² – [3(x – 1)]²

= [2(xy + 1) – 3(x – 1)] [2(xy + 1) + 3(x – 1)]

= (2xy + 2 – 3x + 3) (2xy + 2 + 3x – 3)

= (2xy – 3x + 5) (2xy + 3x – 1)

Question 37. (2x + 1)² – 9x⁴

Solution:

(2x + 1)² – 9x⁴

= (2x + 1)² – (3x² )²

= [(2x +1) – 3x²] [(2x + 1) + 3x²]

= (-3x² + 2x + 1)(3x² + 2x + 1)

= (-3x² + 3x – x + 1) (3x² + 2x + 1)

= [3x(1 – x) + (1 – x)] (3x² + 2x + 1)

= (3x +1) (1 – x) (3x² + 2x + 1)

Question 38. x⁴ – (2y – 3z)²

Solution:

x⁴ – (2y – 3z)²

= (x²)² – (2y – 3z)²

= [x² – (2y – 3z)] [x² + (2y – 3z)]

= (x² – 2y + 3z) (x² + 2y – 3z)

Question 39. a² – b² + a – b

Solution:

a² – b² + a – b

= (a² – b²) + (a – b)

= [(a – b) (a + b)] + (a – b)

= (a – b) (a + b + 1)

Question 40. 16a⁴ – b⁴

Solution:

16a⁴ – b⁴

= (4a²)² – (b²)²

= (4a² – b²) (4a² + b²)

= [(2a)² – b²] (4a²+ b²)

= (2a – b) (2a + b) (4a² + b²)

Question 41. a⁴ – 16(b – c)⁴

Solution:

a⁴ – 16(b – c)⁴

= (a²)² – [4(b – c)²]²

= [a² – 4(b – c)²] [a² + 4(b – c)²]

= {a² – [2(b – c)]²}[a² + 4(b – c)²]

= [a – 2(b – c)] [a + 2(b – c)] [a² + 4(b – c)²]

= (a – 2b + 2c) (a + 2b – 2c) [a² + 4(b – c)²]

Question 42. 2a⁵ – 32a

Solution:

2a⁵ – 32a

= 2a(a⁵ – 1 )

= 2a [ (a²)² – 1² ]

= 2a ( a2 – 1) (a2+ 1)

= 2a (a+1) (a-1) (a²+1)

Question 43. a⁴b⁴ – 81c⁴

Solution:

Answer:

   a⁴b⁴ – 81c⁴

= (a²b²)² – (9c²)²

= (a²b² – 9c²) (a²b² + 9c²)

= [(ab)² – (3c)² ] (a²b² + 9c²)

= (ab – 3c) (ab + 3c) (a²b² + 9c²)

Question 44. xy⁹ – x⁹y

Solution:

xy⁹ – x⁹y

= xy(y⁸ – x⁸)

= xy [(y⁴)² – (x⁴)²]

= xy (y⁴ – x⁴)(y⁴ + x⁴)

= xy [(y²)² – (x²)²] (y⁴ + x⁴)

= xy [(y² – x²)(y² + x²)(y⁴ + x⁴)

= xy (y – x) (y + x) (y² + x²) (y⁴ + x⁴)

Question 45. x³ – x

Solution:

x³ – x

= x(x² – 1)

= x (x + 1)(x – 1)

Question 46. 18a²x² – 32

Solution:

18a²x² – 32

= 2(9a²x² – 16)

= 2[(3ax)² – 42]

= 2(3ax – 4) (3ax + 4)