8类RD Sharma解决方案–第7章分解–练习7.5 |套装1

分解以下每个表达式：

问题1. 16x ² – 25y ²

解决方案：

16x² – 25y² 

= (4x)² – (5y)² 

= (4x – 5y)(4x + 5y)  

问题2：27x ² – 12y ²

解决方案：

27x² – 12y²

= 3(9x² – 4y²)

= 3[(3x)² – (2y)²]

= 3(3x – 2y)(3x + 2y) 

问题3. 144a ² – 289b ²

解决方案：

144a² – 289b²

= (12a)² – (17b)² 

= (12a – 17b)(12a + 17b) 

问题4. 12m ² – 27

解决方案：

12m² – 27

= 3(4m² – 9) 

= 3[(2m)² – 3²] 

= 3(2m – 3)(2m + 3) 

问题5. 125x ² – 45y ²

解决方案：

125x² – 45y²

= 5(25x² – 9y²) 

= 5[(5x)² – (3y)²] 

= 5(5x – 3y)(5x + 3y) 

问题6. 144a ² – 169b ²

解决方案：

144a² – 169b²

= (12a)² – (13b)² 

= (12a – 13b)(12a + 13b)

问题7.(2a – b) ² – 16c ²

解决方案：

(2a – b)² – 16c²

= (2a – b)² – (4c)² 

= [(2a – b) + 4c][(2a – b) – 4c] 

= (2a – b + 4c)(2a – b – 4c) 

问题8.(x + 2y) ² – 4(2x – y) ²

解决方案：

(x + 2y)² – 4(2x – y)²

= (x + 2y)² – [2(2x – y)]² 

= [(x + 2y) – 2(2x – y)][(x + 2y) + 2(2x – y)] 

= (x + 2y – 4x + 2y)(x + 2y + 4x – 2y) 

= 5x(4y – 3x) 

问题9. 3a ⁵ – 48a ³

解决方案：

3a⁵ – 48a³

= 3a³(a² – 16) 

= 3a³(a² – (4)²) 

= 3a³(a – 4)(a + 4) 

问题10. a ⁴ – 16b ⁴

解决方案：

a⁴ – 16b⁴  

= a⁴ – 2⁴b⁴ 

= (a²)² – (2²b²)²

= (a² – 2²b²)(a² + 2²b²) 

= [a² – (2b)²](a² + 4b²) 

= [a² – (2b)²](a² + 4b²) 

= (a – 2b)(a + 2b)(a² + 4b²) 

问题11. x ⁸ – 1

解决方案：

x⁸ – 1

= (x⁴)² – 1²

= [(x²)² – 1²](x⁴ + 1)

= [(x² – 1)(x² + 1)](x⁴ + 1)

= (x – 1)(x + 1)(x² + 1)(x⁴ + 1) 

问题12：64 –(a + 1) ²

解决方案：

64 – (a + 1)²

= (8)² – (a + 1)²

= [8 – (a + 1)][8 + (a + 1)]  

= (8 – a – 1)(8 + a + 1)  

= (7-a)(9+a)  

问题13：36L ² –(m + n) ²

解决方案：

36L² – (m + n)²

= (6L)² – (m + n)²

= [6L – (m + n)][6L + (m + n)]  

= (6L – m – n)(6L + m + n)  

问题14. 25x ^4年4 – 1

解决方案：

25x⁴y⁴ – 1

= (5x²y²)² – 1²

= (5x²y² – 1)(5x²y² + 1)  

问题15. a ⁴ – 1 / b ⁴

解决方案：

a⁴ – 1/b⁴

= (a²)² – (1/b²)²

= (a² – 1/b²)(a² + 1/b²)  

= (a – 1/b)(a + 1/b)(a² + 1/b²)  

问题16. x ³ – 144x

解决方案：

x³ – 144x

= x(x² – 144)  

= x(x² – 12²)  

= x(x – 12)(x + 12)  

问题17.(x – 4y) ² – 625

解决方案：

(x – 4y)² – 625  

= (x – 4y)² – 25²  

= [(x – 4y) – 25] [(x – 4y) + 25]  

= (x – 4y – 25)(x – 4y + 25)

问题18。9(a – b) ² – 100(x – y) ²

解决方案：

9(a – b)² – 100(x – y)²

= [3(a – b)]² – [10 (x – y)]²

= [3(a – b) -10(x – y)] [3(a – b) + 10(x – y)]

= (3a – 3b – 10x + 10y) (3a – 3b + 10x – 10y)

问题19.(3 + 2a) ² – 25a ²

解决方案：

(3 + 2a)² – 25a²

= (3 + 2a)² – (5a)²

= (3 + 2a + 5a)(3 + 2a – 5a)

= (3 + 7a)(3 – 3a)

问题20.(x + y) ² –(a – b) ²

解决方案：

(x + y)² – (a – b)²

= [(x + y) + (a – b)] [(x + y) – (a – b)]

= (x + y + a – b) (x + y – a + b)

问题21. 1/16 x ² y ² – 4 / 49y ² z ²

解决方案：

1/16 x²y² – 4/49y²z²

= (1/4xy)² – (2/7 yz)²

= (1/4xy – 2/7yz) (1/4xy + 2/7yz)

= [y (1/4x – 2/7z)] [y(1/4x + 2/7z)]

= y² (1/4x – 2/7z) (1/4x + 2/7z)

问题22. 75a ³ b ² – 108ab ⁴

解决方案：

75a³b² – 108ab⁴

= 3ab²(25a² – 36b²)

= 3ab²[(5a)² – (6b)²]

= 3ab²[(5a – 6b)(5a + 6b)]

= 3ab²(5a – 6b)(5a + 6b)

问题23. x ⁵ – 16x ³

解决方案：

x⁵ – 16x³

= x³(x² – 16)

= x³(x² – 4²)

= x³(x – 4)(x + 4)

第7章分解-练习7.5 |套装2