Class 8 RD Sharma Solutions β Chapter 6 Algebraic Expressions And Identities β Exercise 6.5 | Set 2
Find the following products and verify the result for x = -1 and y = -2:
Question 17. (3x β 5y) (x + y)
Solution:
To find the product of (3x β 5y) (x + y) [use distributive property]
β 3x (x + y) -5y (x + y)
β 3x2 + 3xy -5xy β 5y2
β 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 β 5y) (x + y)
β [3(-1) -5(-2)] [(-1) + (-2)]
β (-3 +10) (-3)
β (7) (-3)
β -21
RHS: 3x2 -2xy -5y2
β 3 (-1)2 -2 (-1)(-2) -5(-2)2
β 3(1) -2(2) -5(4)
β 3-4 -20
β -21
Therefore, LHS = RHS
Hence, result verified.
Question 18. (x2y β 1) (3 β 2x2y)
Solution:
To find the product of (x2y β 1) (3 β 2x2y) [use distributive property]
β x2y (3 β 2x2y) -1(3 β 2x2y)
β 3x2y -2x4y2 -3 + 2x2y
β 5x2y -2x4y2 β 3
The required product is 5x2y -2x4y2 β 3
Now, putting the values of x = -1 and y = -2 on both the sides and verifying the results .
LHS: (x2y β 1) (3 β 2x2y)
β [(-1)2(-2) β 1] [3 β 2(-1)2(-2)]
β [(1)(-2) -1] [3 β 2(1)(-2)]
β (-3)(7)
β -21
RHS: 5x2y -2x4y2 β 3
β 5 (-1)2(-2) β 2 (-1)4(-2)2 β 3
β 5 (-2) -2(4) β 3
β -10 -8 β 3
β -21
Therefore, LHS = RHS
Hence, result verified.
Question 19. (1/3X β Y2/5) (1/3X + Y2/5)
Solution:
To find the product of (1/3X β Y2/5) (1/3X + Y2/5) [use Identity (a-b)(a+b) = a2 β b2]
β [(1/3X)2 β (Y2/5)2]
β 1/9X2 β Y4 / 25
The required product is 1/9X2 β Y4 / 25
Now, putting the values of x = -1 and y = -2 on both the sides and verifying the results .
LHS: (1/3X β Y2/5) (1/3X + Y2/5)
β [1/3(-1) β (-2)2/5] [1/3 (-1) + (-2)2 / 5]
β [-1/3 -4/5] [-1/3 + 4/5]
β (- 17/15) (7/15)
β -119/225
RHS: 1/9X2 β Y4/ 25
β 1/9 (-1)2 β (-2)4/25
β 1/9 β 16/25
β -119/225
Therefore, LHS = RHS
Hence, result verified.
Simplify:
Question 20. x2 (x + 2y) (x -3y)
Solution:
To find the product of x2 (x + 2y) (x -3y) [use distributive property]
β [x2 (x + 2y)] (x -3y)
β (x3 + 2x2y) (x -3y)
β x3(x -3y) + 2x2y(x -3y)
β x4 -3x3y + 2x3y β 6x2y2
β x4 -x3y -6x2y2
Hence, the required answer is x4 -x3y -6x2y2
Question 21. (x2 β 2y2) (x + 4y) x2y2
Solution:
To find the product of (x2 -2y2) (x+4y) x2y2 [use distributive property]
β [x2 (x + 4y) -2y2 (x+4y)] x2y2
β (x3 + 4x2y -2xy2 -8y3) x2y2
β x5y2 + 4x4y3 -2x3y4 β 8x2y5
Hence, the required answer is x5y2 + 4x4y3 -2x3y4 -8x2y5
Question 22. a2b2 (a + 2b) (3a + b)
Solution:
To find the product of a2b2 (a+2b) (3a + b) [use distributive property]
β a2b2 [a (3a + b)+2b (3a + b)]
β a2b2 (3a2 + ab + 6ab + 2b2)
β 3a4b2 + a3b3 + 6a3b3 + 2a2b4
β 3a4b2 + 7a3b3 + 2a2b4
Hence, the required answer is 3a4b2 + 7a3b3 + 2a2b4
Question 23. x2 (x β y) y2 (x+2y)
Solution:
To find the product of x2 (x β y) y2 (x+2y) [use distributive property]
β x2y2 [x (x+2y) β y (x+2y)]
β x2y2 (x2 + 2xy β xy -2y2)
β x4y2 + 2x3y3 β x3y3 -2x2y4
β x4y2 + x3y3 -2x2y4
Hence, the required answer is x4y2 + x3y3 -2x2y4
Question 24. (x3 -2x2 + 5x -7) (2x β 3)
Solution:
To find the product of (x3 -2x2 + 5x -7) (2x β 3) [use distributive property]
β 2x (x3 -2x2 + 5x -7) -3 (x3 -2x2 + 5x -7)
β 2x4 -4x3 + 5x2 β 14x β 3x3 + 6x2 β 15x -21
β 2x4 β 7x3 +11x2 -29x -21
Hence, the required answer is 2x4 β 7x3 +11x2 -29x -21
Question 25. (5x + 3) (6 β 5x) (2 β x)
Solution:
To find the product of (5x + 3) (6 -5x) (2 β x) [use distributive property]
β [5x (6 -5x) + 3 (6 -5x)] (2-x)
β (30x -25x2 + 18 -15x) (2 β x)
β (-25x2 -5x +18) (2 β x)
β 2 (-25x2 -5x +18) -x (-25x2 -5x +18)
β -50x2 -10x + 36 + 25x3 + 5x2 β 18x
β 25x3 β 45x2 -28x +36
Hence, the required answer is 25x3 β 45x2 -28x + 36
Question 26. (5 β x) (6 β 5x) (2 β x)
Solution:
To find the product of (5 β x) (6 β 5x) (2 β x) [use distributive property]
β [5(6 β 5x) β x(6 β 5x)] (2 β x)
β (30 β 25x β 6x + 5x2) (2 β x)
β (5x2 β 31x + 30) (2 β x)
β 2(5x2 β 31x + 30) β x(5x2 β 31x + 30)
β 10x2 β 62x + 60 β 5x3 + 31x2 β 30x
β -5x3 + 41x2 β 92x + 60
Hence, the required answer is -5x3 + 41x2 β 92x + 60
Question 27. (2x2 + 3x β 5) (3x2 -5x + 4)
Solution:
To find the product of (2x2 + 3x β 5) (3x2 β 5x + 4) [use distributive property]
β [2x2 (3x2 β 5x + 4) + 3x(3x2 β 5x + 4) β 5(3x2 β 5x + 4)]
β (6x4 β 10x3 + 8x2) + (9x3 β 15x2 + 12x) + (- 15x2 + 25x β 20)
β 6x4 β x3 β 22x2 + 37x β 20
Hence, the required answer is 6x4 β x3 β 22x2 + 37x β 20
Question 28. (3x β 2) (2x β 3) + (5x β 3) (x + 1)
Solution:
To find the product of (3x β 2) (2x β 3) + (5x β 3) (x + 1) [use distributive property]
β [3x (2x β 3) β 2(2x β 3)] + [5x(x + 1) -3(x + 1)]
β (6x2 β 9x β 4x + 6) + (5x2 + 5x β 3x β 3)
β 11x2 β 11x + 3
Hence, the required answer is 11x2 β 11x + 3
Question 29. (5x β 3)(x + 2) β (2x + 5) (4x β 3)
Solution:
To find the product of (5x β 3)(x + 2) β (2x + 5) (4x β 3) [use distributive property]
β [5x (x + 2) β 3 (x + 2)] β [2x (4x β 3) + 5(4x β 3)]
β (5x2 + 10 x β 3x β 6) β (8x2 β 6x + 20x -15)
β (5x2 + 7x β 6) β (8x2 + 14x β 15)
β -3x2 β 7x + 9
Hence, the required answer is -3x2 β 7x + 9
Question 30. (3x + 2y) (4x + 3y) β (2x β y) (7x β 3y)
Solution:
To find the product of (3x + 2y) (4x + 3y) β (2x β y) (7x β 3y) [use distributive property]
β [3x (4x + 3y) + 2y (4x + 3y)] β [2x(7x β 3y) β y(7x β 3y)]
β (12x2 + 9xy + 8xy + 6y2) β (14x2 β 6xy β 7xy + 3y2)
β (12x2 + 17xy + 6y2) β (14x2 β 13xy + 3y2)
β -2x2 + 30xy + 3y2
Hence, the required answer is -2x2 + 30xy + 3y2
Question 31. (x2 β 3x + 2) (5x β 2) β (3x2 + 4x β 5)(2x β 1)
Solution:
To find the product of (x2 β 3x + 2) (5x β 2) β (3x2 + 4x β 5)(2x β 1) [use distributive property]
β [5x (x2 β 3x + 2) β 2(x2 β 3x + 2)] β [2x (3x2 + 4x β 5) β 1(3x2 + 4x β 5)]
β (5x3 -15x2 + 10x -2x2 + 6x -4) β (6x3 + 8x2 -10x β 3x2 β 4x + 5)
β (5x3 -17x2 +16x β 4) β (6x3 + 5x2 β 14x + 5)
β -x3 -22x2 + 30x β 9
Hence, the required answer is -x3 β 22x2 + 30x β 9
Question 32. (x3 β 2x2 + 3x β 4)(x -1) β (2x β 1) (x2 β x + 1)
Solution:
To find the product of (x3 β 2x2 + 3x β 4)(x -1) β (2x β 1) (x2 β x + 1) [use distributive property]
β [x(x3 -2x2 +3x -4) β 1(x3 β 2x2 + 3x β 4)] β [2x (x2 β x + 1) β 1(x2 β x + 1)]
β (x4 -2x3 + 3x2 -4x -x3 + 2x2 β 3x + 4) β (2x3 β 2x2 + 2x β x2 + x β 1)
β (x4 -3x3 + 5x2 -7x + 4) β (2x3 -3x2 + 3x -1)
β x4 β 5x3 + 8x2 β 10x + 5
Hence, the required answer is x4 β 5x3 + 8x2 β 10x + 5