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“Factorization of binomial expressions expressible as the difference of two squares” is discussed in Exercise 7.5 of Chapter 7 Factorization.
RD Sharma Solutions for Class 8 Maths Exercise 7.5 Chapter 7 Factorization
Access Answers to RD Sharma Solutions for Class 8 Maths Exercise 7.5 Chapter 7 Factorization
EXERCISE 7.5 PAGE NO: 7.17
Factorize each of the following expressions:
1. 16x2 – 25y2
Solution:
We have,
16x2 – 25y2
(4x)2 – (5y)2
By using the formula (a2 – b2) = (a + b) (a – b), we get,
(4x + 5y) (4x – 5y)
2. 27x2 – 12y2
Solution:
We have,
27x2 – 12y2
By taking 3 as common, we get,
3 [(3x)2 – (2y)2]
By using the formula (a2 – b2) = (a-b) (a+b)
3 (3x + 2y) (3x – 2y)
3. 144a2 – 289b2
Solution:
We have,
144a2 – 289b2
(12a)2 – (17b)2
By using the formula (a2 – b2) = (a-b) (a+b)
(12a + 17b) (12a – 17b)
4. 12m2 – 27
Solution:
We have,
12m2 – 27
By taking 3 as common, we get,
3 (4m2 – 9)
3 [(2m)2 – 32]
By using the formula (a2 – b2) = (a-b) (a+b)
3 (2m + 3) (2m – 3)
5. 125x2 – 45y2
Solution:
We have,
125x2 – 45y2
By taking 5 as common, we get,
5 (25x2 – 9y2)
5 [(5x)2 – (3y)2]
By using the formula (a2 – b2) = (a-b) (a+b)
5 (5x + 3y) (5x – 3y)
6. 144a2 – 169b2
Solution:
We have,
144a2 – 169b2
(12a)2 – (13b)2
By using the formula (a2 – b2) = (a-b) (a+b)
(12a + 13b) (12a – 13b)
7. (2a – b)2 – 16c2
Solution:
We have,
(2a – b)2 – 16c2
(2a – b)2 – (4c)2
By using the formula (a2 – b2) = (a-b) (a+b)
(2a – b + 4c) (2a – b – 4c)
8. (x + 2y)2 – 4 (2x – y)2
Solution:
We have,
(x + 2y)2 – 4 (2x – y)2
(x + 2y)2 – [2 (2x – y)]2
By using the formula (a2 – b2)= (a + b) (a – b), we get,
[(x + 2y) + 2 (2x – y)] [x + 2y – 2 (2x – y)](x + 4x + 2y – 2y) (x – 4x + 2y + 2y)
(5x) (4y – 3x)
9. 3a5 – 48a3
Solution:
We have,
3a5 – 48a3
By taking 3 as common, we get,
3a3 (a2 – 16)
3a3 (a2 – 42)
By using the formula (a2 – b2) = (a-b) (a+b)
3a3 (a + 4) (a – 4)
10. a4 – 16b4
Solution:
We have,
a4 – 16b4
(a2)2 – (4b2)2
By using the formula (a2 – b2) = (a-b) (a+b)
(a2 + 4b2) (a2 – 4b2)
By using the formula (a2 – b2) = (a-b) (a+b)
(a2 + 4b2) (a + 2b) (a – 2b)
11. x8 – 1
Solution:
We have,
x8 – 1
(x4)2–(1)2
By using the formula (a2 – b2) = (a-b) (a+b)
(x4 + 1) (x4 – 1)
By using the formula (a2 – b2) = (a-b) (a+b)
(x4 + 1) (x2 + 1) (x – 1) (x + 1)
12. 64 – (a + 1)2
Solution:
We have,
64 – (a + 1)2
82 – (a + 1)2
By using the formula (a2 – b2) = (a-b) (a+b)
[8 + (a + 1)] [8 – (a + 1)](a + 9) (7 – a)
13. 36l2 – (m + n)2
Solution:
We have,
36l2 – (m + n)2
(6l)2 – (m + n)2
By using the formula (a2 – b2) = (a-b) (a+b)
(6l + m + n) (6l – m – n)
14. 25x4y4 – 1
Solution:
We have,
25x4y4 – 1
(5x2y2)2 – (1)2
By using the formula (a2 – b2) = (a-b) (a+b)
(5x2y2 – 1) (5x2y2 + 1)
15. a4 – 1/b4
Solution:
We have,
a4 – 1/b4
(a2)2 – (1/b2)2
By using the formula (a2 – b2) = (a-b) (a+b)
(a2 + 1/b2) (a2 – 1/b2)
By using the formula (a2 – b2) = (a-b) (a+b)
(a2 + 1/b2) (a – 1/b) (a + 1/b)
16. x3 – 144x
Solution:
We have,
x3 – 144x
x [x2 – (12)2]
By using the formula (a2 – b2) = (a-b) (a+b)
x (x + 12) (x – 12)
17. (x – 4y)2 – 625
Solution:
We have,
(x – 4y)2 – 625
(x – 4y)2 – (25)2
By using the formula (a2 – b2) = (a-b) (a+b)
(x – 4y + 25) (x – 4y – 25)
18. 9 (a – b)2 – 100 (x – y)2
Solution:
We have,
9 (a – b)2 – 100 (x – y)2
[3 (a – b)]2 – [10 (x – y)]2By using the formula (a2 – b2) = (a-b) (a+b)
[3 (a – b) + 10 (x + y)] [3 (a – b) – 10 (x – y)] [3a – 3b + 10x – 10y] [3a – 3b – 10x + 10y]19. (3 + 2a)2 – 25a2
Solution:
We have,
(3 + 2a)2 – 25a2
(3 + 2a)2 – (5a)2
By using the formula (a2 – b2) = (a-b) (a+b)
(3 + 2a + 5a) (3 + 2a – 5a)
(3 + 7a) (3 – 3a)
(3 + 7a) 3(1 – a)
20. (x + y)2 – (a – b)2
Solution:
We have,
(x + y)2 – (a – b)2
By using the formula (a2 – b2) = (a-b) (a+b)
[(x + y) + (a – b)] [(x + y) – (a – b)](x + y + a – b) (x + y – a + b)
21. 1/16x2y2 – 4/49y2z2
Solution:
We have,
1/16x2y2 – 4/49y2z2
(1/4xy)2 – (2/7yz)2
By using the formula (a2 – b2) = (a-b) (a+b)
(xy/4 + 2yz/7) (xy/4 – 2yz/7)
y2 (x/4 + 2/7z) (x/4 – 2/7z)
22. 75a3b2 – 108ab4
Solution:
We have,
75a3b2 – 108ab4
3ab2 (25a2 – 36b2)
3ab2 [(5a)2 – (6b)2]
By using the formula (a2 – b2) = (a-b) (a+b)
3ab2 (5a + 6b) (5a – 6b)
23. x5 – 16x3
Solution:
We have,
x5 – 16x3
x3 (x2 – 16)
x3 (x2 – 42)
By using the formula (a2 – b2) = (a-b) (a+b)
x3 (x + 4) (x – 4)
24. 50/x2 – 2x2/81
Solution:
We have,
50/x2 – 2x2/81
2 (25/x2 – x2/81)
2 [(5/x)2 – (x/9)2]
By using the formula (a2 – b2) = (a-b) (a+b)
2 (5/x+ x/9) (5/x – x/9)
25. 256x3 – 81x
Solution:
We have,
256x3 – 81x
x (256x4 – 81)
x [(16x2)2 – 92]
By using the formula (a2 – b2) = (a-b) (a+b)
x (4x + 3) (4x – 3) (16x2 + 9)
26. a4 – (2b + c)4
Solution:
We have,
a4 – (2b + c)4
(a2)2 – [(2b + c)2]2
By using the formula (a2 – b2) = (a-b) (a+b)
[a2 + (2b + c)2] [a2 – (2b + c)2]By using the formula (a2 – b2) = (a-b) (a+b)
[a2 + (2b + c)2] [a + 2b + c] [a – 2b – c]27. (3x + 4y)4 – x4
Solution:
We have,
(3x + 4y)4 – x4
[(3x + 4y)2]2 – (x2)2By using the formula (a2 – b2) = (a-b) (a+b)
[(3x + 4y)2 + x2] [(3x + 4y)2 – x2] [(3x + 4y)2 + x2] [3x + 4y + x] [3x + 4y – x] [(3x + 4y)2 + x2] [4x + 4y] [2x + 4y] [(3x + 4y)2 + x2] 8[x + 2y] [x + y]28. p2q2 – p4q4
Solution:
We have,
p2q2 – p4q4
(pq)2 – (p2q2)2
By using the formula (a2 – b2) = (a-b) (a+b)
(pq + p2q2) (pq – p2q2)
p2q2 (1 + pq) (1 – pq)
29. 3x3y – 243xy3
Solution:
We have,
3x3y – 243xy3
3xy (x2 – 81y2)
3xy [x2 – (9y)2]
By using the formula (a2 – b2) = (a-b) (a+b)
(3xy) (x + 9y) (x – 9y)
30. a4b4 – 16c4
Solution:
We have,
a4b4 – 16c4
(a2b2)2 – (4c2)2
By using the formula (a2 – b2) = (a-b) (a+b)
(a2b2 + 4c2) (a2b2 – 4c2)
By using the formula (a2 – b2) = (a-b) (a+b)
(a2b2 + 4c2) (ab + 2c) (ab – 2c)
31. x4 – 625
Solution:
We have,
x4 – 625
(x2)2 – (25)2
By using the formula (a2 – b2) = (a-b) (a+b)
(x2 + 25) (x2 – 25)
(x2 + 25) (x2 – 52)
By using the formula (a2 – b2) = (a-b) (a+b)
(x2 + 25) (x + 5) (x – 5)
32. x4 – 1
Solution:
We have,
x4 – 1
(x2)2 – (1)2
By using the formula (a2 – b2) = (a-b) (a+b)
(x2 + 1) (x2 – 1)
By using the formula (a2 – b2) = (a-b) (a+b)
(x2 + 1) (x + 1) (x – 1)
33. 49(a – b)2 – 25(a + b)2
Solution:
We have,
49(a – b)2 – 25(a + b)2
[7 (a – b)]2 – [5 (a + b)]2By using the formula (a2 – b2) = (a-b) (a+b)
[7 (a – b) + 5 (a + b)] [7 (a – b) – 5 (a + b)](7a – 7b + 5a + 5b) (7a – 7b – 5a – 5b)
(12a – 2b) (2a – 12b)
2 (6a – b) 2 (a – 6b)
4 (6a – b) (a – 6b)
34. x – y – x2 + y2
Solution:
We have,
x – y – x2 + y2
x – y – (x2 – y2)
By using the formula (a2 – b2) = (a-b) (a+b)
x – y – (x + y) (x – y)
(x – y) (1 – x – y)
35. 16(2x – 1)2 – 25y2
Solution:
We have,
16(2x – 1)2 – 25y2
[4 (2x – 1)]2 – (5y)2By using the formula (a2 – b2) = (a-b) (a+b)
(8x + 5y – 4) (8x – 5y – 4)
36. 4(xy + 1)2 – 9(x – 1)2
Solution:
We have,
4(xy + 1)2 – 9(x – 1)2
[2 (xy + 1)]2 – [3 (x – 1)]2By using the formula (a2 – b2) = (a-b) (a+b)
(2xy + 2 + 3x – 3) (2xy + 2 – 3x + 3)
(2xy + 3x – 1) (2xy – 3x + 5)
37. (2x + 1)2 – 9x4
Solution:
We have,
(2x + 1)2 – 9x4
(2x + 1)2 – (3x2)2
By using the formula (a2 – b2) = (a-b) (a+b)
(2x + 1 + 3x2) (2x + 1 – 3x2)
(3x2 + 2x + 1) (-3x2 + 2x + 1)
38. x4 – (2y – 3z)2
Solution:
We have,
x4 – (2y – 3z)2
(x2)2 – (2y – 3z)2
By using the formula (a2 – b2) = (a-b) (a+b)
(x2 + 2y – 3z) (x2 – 2y + 3z)
39. a2 – b2 + a – b
Solution:
We have,
a2 – b2 + a – b
By using the formula (a2 – b2) = (a-b) (a+b)
(a + b) (a – b) + (a – b)
(a – b) (a + b + 1)
40. 16a4 – b4
Solution:
We have,
16a4 – b4
(4a2)2 – (b2)2
(4a2 + b2) (4a2 – b2)
By using the formula (a2 – b2) = (a-b) (a+b)
(4a2 + b2) (2a + b) (2a – b)
41. a4 – 16(b – c)4
Solution:
We have,
a4 – 16(b – c)4
(a2)2 – [4 (b – c)2]
By using the formula (a2 – b2) = (a-b) (a+b)
[a2 + 4 (b – c)2] [a2 – 4 (b – c)2]By using the formula (a2 – b2) = (a-b) (a+b)
[a2 + 4 (b – c)2] [(a + 2b – 2c) (a – 2b + 2c)]42. 2a5 – 32a
Solution:
We have,
2a5 – 32a
2a (a4 – 16)
2a [(a2)2 – (4)2]
By using the formula (a2 – b2) = (a-b) (a+b)
2a (a2 + 4) (a2 – 4)
2a (a2 + 4) (a2 – 22)
By using the formula (a2 – b2) = (a-b) (a+b)
2a (a2 + 4) (a + 2) (a – 2)
43. a4b4 – 81c4
Solution:
We have,
a4b4 – 81c4
(a2b2)2 – (9c2)2
By using the formula (a2 – b2) = (a-b) (a+b)
(a2b2 + 9c2) (a2b2 – 9c2)
By using the formula (a2 – b2) = (a-b) (a+b)
(a2b2 + 9c2) (ab + 3c) (ab – 3c)
44. xy9 – yx9
Solution:
We have,
xy9 – yx9
-xy (x8 – y8)
-xy [(x4)2 – (y4)2]
By using the formula (a2 – b2) = (a-b) (a+b)
-xy (x4 + y4) (x4 – y4)
By using the formula (a2 – b2) = (a-b) (a+b)
-xy (x4 + y4) (x2 + y2) (x2 – y2)
By using the formula (a2 – b2) = (a-b) (a+b)
-xy (x4 + y4) (x2 + y2) (x + y) (x – y)
45. x3 – x
Solution:
We have,
x3 – x
x (x2 – 1)
By using the formula (a2 – b2) = (a-b) (a+b)
x (x + 1) (x – 1)
46. 18a2x2 – 32
Solution:
We have,
18a2x2 – 32
2 [(3ax)2 – (4)2]
By using the formula (a2 – b2) = (a-b) (a+b)
2 (3ax + 4) (3ax – 4)
Thanks sir it is so helpful