第6章代数表达式和恒等式–练习6.5 |套装1
查找以下产品，并验证x = -1和y = -2的结果：
问题17.(3x – 5y)(x + y)
解决方案：
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.
问题18.(x 2 y – 1)(3 – 2x 2 y)
解决方案：
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.
问题19(1 / 3X – Y 2/5)(1 / 3X + Y 2/5)
解决方案：
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.
简化：
问题20.x 2 (x + 2y)(x -3y)
解决方案：
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
问题21.(x 2 – 2y 2 )(x + 4y)x 2 y 2
解决方案：
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
问题22. a 2 b 2 (a + 2b)(3a + b)
解决方案：
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
问题23. x 2 (x – y)y 2 (x + 2y)
解决方案：
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
问题24.(x 3 -2x 2 + 5x -7)(2x – 3)
解决方案：
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
问题25.(5x + 3)(6 – 5x)(2 – x)
解决方案：
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
问题26.(5 – x)(6 – 5x)(2 – x)
解决方案：
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
问题27.(2x 2 + 3x – 5)(3x 2 -5x + 4)
解决方案：
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
问题28.(3x – 2)(2x – 3)+(5x – 3)(x + 1)
解决方案：
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
问题29.(5x – 3)(x + 2)–(2x + 5)(4x – 3)
解决方案：
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
问题30。(3x + 2y)(4x + 3y)–(2x – y)(7x – 3y)
解决方案：
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
问题31.(x 2 – 3x + 2)(5x – 2)–(3x 2 + 4x – 5)(2x – 1)
解决方案：
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
问题32.(x 3 – 2x 2 + 3x – 4)(x -1)–(2x – 1)(x 2 – x + 1)
解决方案：
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