Class 8 RD Sharma Solutions Chapter 6 Algebraic Expressions And Identities Exercise 6.5 | Set 2 (original) (raw)
Last Updated : 23 Jul, 2025
In this exercise, we explore the multiplication of algebraic expressions using distributive properties and identities. Each problem involves finding the product of given expressions and verifying the results by substituting specific values for the variables. This practice reinforces the understanding of algebraic manipulation and helps in simplifying complex expressions.
**Find the following products and verify the result for x = -1 and y = -2:
**Question 17. (3x - 5y) (x + y)
**Solution:
To find the product of (3x - 5y) (x + y) [use distributive property]
⇒ 3x (x + y) -5y (x + y)
⇒ 3x2 + 3xy -5xy - 5y2
⇒ 3x2 -2xy -5y2
The required product is 3x2 -2xy -5y2
Now, putting the values of x = -1 and y = -2 on both the sides and verifying the results .
**LHS: (3x - 5y) (x + y)
⇒ [3(-1) -5(-2)] [(-1) + (-2)]
⇒ (-3 +10) (-3)
⇒ (7) (-3)
⇒ -21
**RHS: 3x2 -2xy -5y2
⇒ 3 (-1)2 -2 (-1)(-2) -5(-2)2
⇒ 3(1) -2(2) -5(4)
⇒ 3-4 -20
⇒ -21
Therefore, LHS = RHS
Hence, result verified.
**Question 18. (x 2 y - 1) (3 - 2x 2 y)
**Solution:
To find the product of (x2y - 1) (3 - 2x2y) **[use distributive property]
⇒ x2y (3 - 2x2y) -1(3 - 2x2y)
⇒ 3x2y -2x4y2 -3 + 2x2y
⇒ 5x2y -2x4y2 - 3
The required product is 5x2y -2x4y2 - 3
Now, putting the values of x = -1 and y = -2 on both the sides and verifying the results .
**LHS: (x2y - 1) (3 - 2x2y)
⇒ [(-1)2(-2) - 1] [3 - 2(-1)2(-2)]
⇒ [(1)(-2) -1] [3 - 2(1)(-2)]
⇒ (-3)(7)
⇒ -21
**RHS: 5x2y -2x4y2 - 3
⇒ 5 (-1)2(-2) - 2 (-1)4(-2)2 - 3
⇒ 5 (-2) -2(4) - 3
⇒ -10 -8 - 3
⇒ -21
Therefore, LHS = RHS
Hence, result verified.
**Question 19. (1/3X - Y 2 /5) (1/3X + Y 2 /5)
**Solution:
To find the product of (1/3X - Y2/5) (1/3X + Y2/5) **[use Identity (a-b)(a+b) = a 2 - b 2 ]
⇒ [(1/3X)2 - (Y2/5)2]
⇒ 1/9X2 - Y4 / 25
The required product is 1/9X2 - Y4 / 25
Now, putting the values of x = -1 and y = -2 on both the sides and verifying the results .
**LHS: (1/3X - Y2/5) (1/3X + Y2/5)
⇒ [1/3(-1) - (-2)2/5] [1/3 (-1) + (-2)2 / 5]
⇒ [-1/3 -4/5] [-1/3 + 4/5]
⇒ (- 17/15) (7/15)
⇒ -119/225
RHS: 1/9X2 - Y4/ 25
⇒ 1/9 (-1)2 - (-2)4/25
⇒ 1/9 - 16/25
⇒ -119/225
Therefore, LHS = RHS
Hence, result verified.
Simplify:
**Question 20. x 2 ****(x + 2y) (x -3y)**
**Solution:
To find the product of x2 (x + 2y) (x -3y) [use distributive property]
⇒ [x2 (x + 2y)] (x -3y)
⇒ (x3 + 2x2y) (x -3y)
⇒ x3(x -3y) + 2x2y(x -3y)
⇒ x4 -3x3y + 2x3y - 6x2y2
⇒ x4 -x3y -6x2y2
Hence, the required answer is x4 -x3y -6x2y2
**Question 21. (x 2 - 2y 2 ) (x + 4y) x 2 y 2
**Solution:
To find the product of (x2 -2y2) (x+4y) x2y2 **[use distributive property]
⇒ [x2 (x + 4y) -2y2 (x+4y)] x2y2
⇒ (x3 + 4x2y -2xy2 -8y3) x2y2
⇒ x5y2 + 4x4y3 -2x3y4 - 8x2y5
Hence, the required answer is x5y2 + 4x4y3 -2x3y4 -8x2y5
**Question 22. a 2 b 2 (a + 2b) (3a + b)
**Solution:
To find the product of a2b2 (a+2b) (3a + b) [use distributive property]
⇒ a2b2 [a (3a + b)+2b (3a + b)]
⇒ a2b2 (3a2 + ab + 6ab + 2b2)
⇒ 3a4b2 + a3b3 + 6a3b3 + 2a2b4
⇒ 3a4b2 + 7a3b3 + 2a2b4
Hence, the required answer is 3a4b2 + 7a3b3 + 2a2b4
**Question 23. x 2 (x - y) y 2 (x+2y)
**Solution:
To find the product of x2 (x - y) y2 (x+2y) **[use distributive property]
⇒ x2y2 [x (x+2y) - y (x+2y)]
⇒ x2y2 (x2 + 2xy - xy -2y2)
⇒ x4y2 + 2x3y3 - x3y3 -2x2y4
⇒ x4y2 + x3y3 -2x2y4
Hence, the required answer is x4y2 + x3y3 -2x2y4
**Question 24. (x 3 -2x 2 + 5x -7) (2x - 3)
**Solution:
To find the product of (x3 -2x2 + 5x -7) (2x - 3) **[use distributive property]
⇒ 2x (x3 -2x2 + 5x -7) -3 (x3 -2x2 + 5x -7)
⇒ 2x4 -4x3 + 5x2 - 14x - 3x3 + 6x2 - 15x -21
⇒ 2x4 - 7x3 +11x2 -29x -21
Hence, the required answer is 2x4 - 7x3 +11x2 -29x -21
**Question 25. (5x + 3) (6 - 5x) (2 - x)
**Solution:
To find the product of (5x + 3) (6 -5x) (2 - x) **[use distributive property]
⇒ [5x (6 -5x) + 3 (6 -5x)] (2-x)
⇒ (30x -25x2 + 18 -15x) (2 - x)
⇒ (-25x2 -5x +18) (2 - x)
⇒ 2 (-25x2 -5x +18) -x (-25x2 -5x +18)
⇒ -50x2 -10x + 36 + 25x3 + 5x2 - 18x
⇒ 25x3 - 45x2 -28x +36
Hence, the required answer is 25x3 - 45x2 -28x + 36
**Question 26. (5 - x) (6 - 5x) (2 - x)
**Solution:
To find the product of (5 - x) (6 - 5x) (2 - x) **[use distributive property]
⇒ [5(6 - 5x) - x(6 - 5x)] (2 - x)
⇒ (30 - 25x - 6x + 5x2) (2 - x)
⇒ (5x2 - 31x + 30) (2 - x)
⇒ 2(5x2 - 31x + 30) - x(5x2 - 31x + 30)
⇒ 10x2 - 62x + 60 - 5x3 + 31x2 - 30x
⇒ -5x3 + 41x2 - 92x + 60
Hence, the required answer is -5x3 + 41x2 - 92x + 60
**Question 27. (2x 2 + 3x - 5) (3x 2 -5x + 4)
**Solution:
To find the product of (2x2 + 3x - 5) (3x2 - 5x + 4) **[use distributive property]
⇒ [2x2 (3x2 - 5x + 4) + 3x(3x2 - 5x + 4) - 5(3x2 - 5x + 4)]
⇒ (6x4 - 10x3 + 8x2) + (9x3 - 15x2 + 12x) + (- 15x2 + 25x - 20)
⇒ 6x4 - x3 - 22x2 + 37x - 20
Hence, the required answer is 6x4 - x3 - 22x2 + 37x - 20
**Question 28. (3x - 2) (2x - 3) + (5x - 3) (x + 1)
**Solution:
To find the product of (3x - 2) (2x - 3) + (5x - 3) (x + 1) **[use distributive property]
⇒ [3x (2x - 3) - 2(2x - 3)] + [5x(x + 1) -3(x + 1)]
⇒ (6x2 - 9x - 4x + 6) + (5x2 + 5x - 3x - 3)
⇒ 11x2 - 11x + 3
Hence, the required answer is 11x2 - 11x + 3
**Question 29. (5x - 3)(x + 2) - (2x + 5) (4x - 3)
**Solution:
To find the product of (5x - 3)(x + 2) - (2x + 5) (4x - 3) **[use distributive property]
⇒ [5x (x + 2) - 3 (x + 2)] - [2x (4x - 3) + 5(4x - 3)]
⇒ (5x2 + 10 x - 3x - 6) - (8x2 - 6x + 20x -15)
⇒ (5x2 + 7x - 6) - (8x2 + 14x - 15)
⇒ -3x2 - 7x + 9
Hence, the required answer is -3x2 - 7x + 9
**Question 30. (3x + 2y) (4x + 3y) - (2x - y) (7x - 3y)
**Solution:
To find the product of (3x + 2y) (4x + 3y) - (2x - y) (7x - 3y) **[use distributive property]
⇒ [3x (4x + 3y) + 2y (4x + 3y)] - [2x(7x - 3y) - y(7x - 3y)]
⇒ (12x2 + 9xy + 8xy + 6y2) - (14x2 - 6xy - 7xy + 3y2)
⇒ (12x2 + 17xy + 6y2) - (14x2 - 13xy + 3y2)
⇒ -2x2 + 30xy + 3y2
Hence, the required answer is -2x2 + 30xy + 3y2
**Question 31. (x 2 - 3x + 2) (5x - 2) - (3x 2 + 4x - 5)(2x - 1)
**Solution:
To find the product of (x2 - 3x + 2) (5x - 2) - (3x2 + 4x - 5)(2x - 1) **[use distributive property]
⇒ [5x (x2 - 3x + 2) - 2(x2 - 3x + 2)] - [2x (3x2 + 4x - 5) - 1(3x2 + 4x - 5)]
⇒ (5x3 -15x2 + 10x -2x2 + 6x -4) - (6x3 + 8x2 -10x - 3x2 - 4x + 5)
⇒ (5x3 -17x2 +16x - 4) - (6x3 + 5x2 - 14x + 5)
⇒ -x3 -22x2 + 30x - 9
Hence, the required answer is -x3 - 22x2 + 30x - 9
**Question 32. (x 3 - 2x 2 + 3x - 4)(x -1) - (2x - 1) (x 2 - x + 1)
**Solution:
To find the product of (x3 - 2x2 + 3x - 4)(x -1) - (2x - 1) (x2 - x + 1) **[use distributive property]
⇒ [x(x3 -2x2 +3x -4) - 1(x3 - 2x2 + 3x - 4)] - [2x (x2 - x + 1) - 1(x2 - x + 1)]
⇒ (x4 -2x3 + 3x2 -4x -x3 + 2x2 - 3x + 4) - (2x3 - 2x2 + 2x - x2 + x - 1)
⇒ (x4 -3x3 + 5x2 -7x + 4) - (2x3 -3x2 + 3x -1)
⇒ x4 - 5x3 + 8x2 - 10x + 5
Hence, the required answer is x4 - 5x3 + 8x2 - 10x + 5
**Also Read: **Chapter 6 Algebraic Expressions And Identities - Exercise 6.5 | Set 1