第7章分解-练习7.5 |套装1
分解以下每个表达式：
问题24 50 /×2 – 2倍81分之2
解决方案：
50/x2 – 2x2/81
= 2(25/x2 – x2/81)
= 2[(5/x)2 – (x/9)2]
= 2(5/x – x/9) (5/x + x/9)
问题25. 256x 5 – 81x
解决方案：
256x5 – 81x
= x(256x4 – 81)
= x[(16x2)2 – (9)2]
= x(16x2 – 9) (16x2 + 9)
问题26. a 4 –(2b + c) 4
解决方案：
a4 – (2b + c)4
= (a2)2 – [(2b + c)2]2
= [a2 – (2b + c)2] [a2 + (2b + c)2]
= [a + (2b + c)] [a – (2b + c)] [a2 + (2b + c)2]
= (a + 2b + c) (a – 2b – c) [a2 + (2b + c)2]
问题27.(3x + 4y) 4 – x 4
解决方案：
(3x + 4y)4 – x4
= [(3x + 4y)2]2 – (x2)2
= [(3x + 4y)2 – x2] [(3x + 4y)2 + x2]
= [(3x + 4y) – x] [(3x + 4y) + x] [(3x + 4y)2 + x2]
= (2x + 4y) (4x + 4y) [(3x + 4y)2 + x2]
= 8(x + 2y) (x + y) [(3x + 4y)2 + x2]
问题28。p 2 q 2 – p 4 q 4
解决方案：
p2q2 – p4q4
= p2q2 (1 – p2q2)
= p2q2 (1 + pq) (1 – pq)
问题29. 3x 3年– 243xy 3
解决方案：
3x3y – 243xy3
= 3xy (x2 – 81y2)
= 3xy [x2 – (9y)2]
= 3xy (x – 9y) (x + 9y)
问题30. a 4 b 4 – 16c 4
解决方案：
a4b4 – 16c4
= (a2b2)2 – (4c2)2
= (a2b2 – 4c2) (a2b2 + c2)
= [a2b2 – (2c)2] (a2b2 + c2)
= (ab – 2c) (ab + 2c) (a2b2 + c2)
问题31. x 4 – 625
解决方案：
x4 – 625
= (x2)2 – (25)2
= (x2 – 25) (x2 + 25)
= (x2 – 52) (x2 + 25)
= (x – 5) (x + 5) (x2 + 25)
问题32. x 4 – 1
解决方案：
x4 – 1
= (x2)2 – 12
= (x2 – 1)(x2 +1)
= (x + 1)(x – 1) (x2 + 1)
问题33. 49(a – b) 2 – 25(a + b) 2
解决方案：
49(a – b)2 – 25(a + b)2
= [7(a – b)]2 – [5 (a + b)]2
= [7(a – b) – 5 (a + b)] [7(a – b) + 5 (a + b)]
= (7a – 7b -5a – 5b) (7a – 7b + 5a + 5b)
= (2a – 12b) (12a – 2b)
= 4(a – 6b)(6a – b)
问题34。x – y – x 2 + y 2
解决方案：
x – y – x2 + y2
= (x – y) – (x2 – y2)
= (x – y) – (x – y)(x + y)
= (x – y) [1 – (x + y)]
= (x – y) (1 – x – y)
问题35. 16(2x – 1) 2 – 25y 2
解决方案：
16(2x – 1)2 – 25y2
= [4(2x – 1)]2 – (5y)2
= [4(2x – 1) – 5y] [4(2x – 1) + 5y]
= (8x – 4 – 5y) ( 8x – 4 + 5y)
问题36. 4(xy +1) 2 – 9(x – 1) 2
解决方案：
4(xy + 1)2 – 9( x- 1)2
= [2(xy + 1)]2 – [3(x – 1)]2
= [2(xy + 1) – 3(x – 1)] [2(xy + 1) + 3(x – 1)]
= (2xy + 2 – 3x + 3) (2xy + 2 + 3x – 3)
= (2xy – 3x + 5) (2xy + 3x – 1)
问题37.(2x +1) 2 – 9x 4
解决方案：
(2x + 1)2 – 9x4
= (2x + 1)2 – (3x2 )2
= [(2x +1) – 3x2] [(2x + 1) + 3x2]
= (-3x2 + 2x + 1)(3x2 + 2x + 1)
= (-3x2 + 3x – x + 1) (3x2 + 2x + 1)
= [3x(1 – x) + (1 – x)] (3x2 + 2x + 1)
= (3x +1) (1 – x) (3x2 + 2x + 1)
问题38。x 4 –(2y – 3z) 2
解决方案：
x4 – (2y – 3z)2
= (x2)2 – (2y – 3z)2
= [x2 – (2y – 3z)] [x2 + (2y – 3z)]
= (x2 – 2y + 3z) (x2 + 2y – 3z)
问题39. a 2 – b 2 + a – b
解决方案：
a2 – b2 + a – b
= (a2 – b2) + (a – b)
= [(a – b) (a + b)] + (a – b)
= (a – b) (a + b + 1)
问题40. 16a 4 – b 4
解决方案：
16a4 – b4
= (4a2)2 – (b2)2
= (4a2 – b2) (4a2 + b2)
= [(2a)2 – b2] (4a2+ b2)
= (2a – b) (2a + b) (4a2 + b2)
问题41. a 4 – 16(b – c) 4
解决方案：
a4 – 16(b – c)4
= (a2)2 – [4(b – c)2]2
= [a2 – 4(b – c)2] [a2 + 4(b – c)2]
= {a2 – [2(b – c)]2}[a2 + 4(b – c)2]
= [a – 2(b – c)] [a + 2(b – c)] [a2 + 4(b – c)2]
= (a – 2b + 2c) (a + 2b – 2c) [a2 + 4(b – c)2]
问题42. 2a 5 – 32a
解决方案：
2a5 – 32a
= 2a(a5 – 1 )
= 2a [ (a2)2 – 12 ]
= 2a ( a2 – 1) (a2+ 1)
= 2a (a+1) (a-1) (a2+1)
问题43. a 4 b 4 – 81c 4
解决方案：
Answer:
a4b4 – 81c4
= (a2b2)2 – (9c2)2
= (a2b2 – 9c2) (a2b2 + 9c2)
= [(ab)2 – (3c)2 ] (a2b2 + 9c2)
= (ab – 3c) (ab + 3c) (a2b2 + 9c2)
问题44. xy 9 – x 9 y
解决方案：
xy9 – x9y
= xy(y8 – x8)
= xy [(y4)2 – (x4)2]
= xy (y4 – x4)(y4 + x4)
= xy [(y2)2 – (x2)2] (y4 + x4)
= xy [(y2 – x2)(y2 + x2)(y4 + x4)
= xy (y – x) (y + x) (y2 + x2) (y4 + x4)
问题45. x 3 – x
解决方案：
x3 – x
= x(x2 – 1)
= x (x + 1)(x – 1)
问题46. 18a 2 x 2 – 32
解决方案：
18a2x2 – 32
= 2(9a2x2 – 16)
= 2[(3ax)2 – 42]
= 2(3ax – 4) (3ax + 4)