Class 8 RD Sharma Solutions - Chapter 6 Algebraic Expressions And Identities - Exercise 6.5 | Set 2

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. (x²y - 1) (3 - 2x²y)

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 - Y²/5) (1/3X + Y²/5)

Solution:

To find the product of (1/3X - Y²/5) (1/3X + Y²/5) [use Identity (a-b)(a+b) = a² - b²]

⇒ [(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. x² (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. (x² - 2y²) (x + 4y) x²y²

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. a²b² (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. x² (x - y) y² (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. (x³ -2x² + 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. (2x² + 3x - 5) (3x² -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. (x² - 3x + 2) (5x - 2) - (3x² + 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. (x³ - 2x² + 3x - 4)(x -1) - (2x - 1) (x² - 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