Chapter 7: Factorization Exercise – 7.5
Question: 1
Solve:
16x2 – 25y2
Solution:
16x2 – 25y2
= (4x)2 – (5y)2
= (4x – 5y)(4x + 5y)
Question: 2
Solve:
27x2 – 12y2
Solution:
27x2 – 12y2
= 3(9x2 – 4y2)
= 3[(3x)2 – (2y)2]
= 3(3x – 2y)(3x + 2y)
Question: 3
Solve:
144a2 – 289b2
Solution:
144a2 – 289b2
= (12a)2 – (17b)2
= (12a – 17b)(12a + 17b)
Question: 4
Solve:
12m2 – 27
Solution:
12m2 – 27
= 3(4m2 – 9)
= 3[(2m)2 – 32]
= 3(2m – 3)(2m + 3)
Question: 5
Solve:
125x2 – 45y2
Solution:
125x2 – 45y2
= 5(25x2 – 9y2)
= 5[(5x)2 – (3y)2]
= 5(5x – 3y)(5x + 3y)
Question: 6
Solve:
144a2 – 169b2
Solution:
144a2 – 169b2
= (12a)2 – (13b2)
= (12a – 13b)(12a + 13b)
Question: 7
Solve:
(2a – b)2 – 16c2
Solution:
(2a – b)2 – 16c2
= (2a – b)2 – (4c)2
= [(2a – b) – 4c][(2a – b) + 4c]
= (2a – b – 4c)(2a – b + 4c)
Question: 8
Solve:
(x + 2y)2 – 4(2x – y)2
Solution:
(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)
Question: 9
Solve:
3a5 – 48a3
Solution:
3a5 – 48a3
= 3a3(a2 – 16)
= 3a3(a2 – 42)
= 3a3(a – 4)(a + 4)
Question: 10
Solve:
a4 – 16b4
Solution:
a4 – 16b4
= a4 – 24b4
= (a2)2 – (22b2)2
= (a2 – 22b2)(a2 + 22b2)
= [a2 – (2b)2](a2 + 4b2)
(a – 2b)(a + 2b)(a2 + 4b2)
Question: 11
Solve:
x8– 1
Solution:
x8– 1 = (x4)2 – 12
= (x4 – 1)(x4 + 1)
= [(x2)2 – 12](x4 + 1)
= (x2 – 1)(x2 + 1)(x4 + 1)
= (x2 – 12)(x2 + 1)(x4 + 1)
= (x – 1)(x +1)(x2 + 1)(x4 + 1)
Question: 12
Solve:
64 – (a + 1)2
Solution:
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)
Question: 13
Solve:
36L2 – (m + n)2
Solution:
36L2 – (m + n)2
= (6L)2 – (m + n)2
= [6L – (m +n)][6L + (m +n)]
= (6L – m – n)(6L + m + n)
Question: 14
Solve:
25x4y4 – 1
Solution:
25x4y4 – 1
= (5x2y2)2 – 1
= (5x2y2 – 1)(5x2y2 + 1)
Question: 15
Solve:
a4 – 1/b4
Solution:
a4 – 1/b4
= (a2)2 – 1/(b2)2
= a2 – 1/b2a2 + 1/b2
= a – 1/ba + 1/ba2 + 1/b2
Question: 16
Solve:
x3 – 144x
Solution:
x3 – 144x
= x(x2 – 144)
= x(x2 – 122)
= x(x – 12)(x + 12)
Question: 17
Solve:
(x – 4y)2 – 625
Solution:
(x – 4y)2 – 625
= (x – 4y)2 – 252
= [(x – 4y) – 25][(x – 4y) + 25]
= (x – 4y – 25)(x – 4y + 25)
Question: 18
Solve:
9(a – b)2 – 100(x – y)2
Solution:
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)
Question: 19
Solve:
(3 + 2a)2 – 25a2
Solution:
(3 + 2a)2 – 25a2
= (3 + 2a)2 – (5a)2
= [(3 + 2a) – 5a][(3 + 2a) + 5a]
= (3 + 2a – 5a)(3 + 2a + 5a)
= (3 – 3a)(3 + 7a)
= 3(1 - a)(3 + 7a)
Question: 20
Solve:
(x + y)2 – (a – b)2
Solution:
(x + y)2 – (a – b)2
= [(x + y) – (a – b)][(x + y) + (a – b)]
= (x + y – a + b)(x + y + a – b)
Question: 21
Solve:
Solution:
Question: 22
Solve:
75a3b2 – 108ab4
Solution:
75a3b2 – 108ab4
= 3ab2(25a2 – 36b2)
= 3ab2 [(5a)2 – (6b)2]
= 3ab2 (5a – 6b) (5a + 6b)
Question: 23
Solve:
x5 – 16x3
Solution:
x5 - 16x3
= x3(x2 – 16)
= x3(x2 – 42)
= x3(x – 4)(x + 4)
Question: 24
Solve:
Solution:
Question: 25
Solve:
256x5 – 81x
Solution:
256x5 – 81x
= x(256x4 – 81)
= x[(16x2)2 – 92]
= x(16x2 + 9)(16x2 – 9)
= x(16x2 + 9)[(4x)2 – 32]
= x(16x2 + 9)(4x + 3)(4x – 3)
Question: 26
Solve:
a4- (2b + c)4
Solution:
a4 – (2b + c)4
= (a2)2 – [(2b + c)2]2
= [a2 + (2b + c)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)
Question: 27
Solve:
(3x + 4y)4 – x4
Solution:
(3x + 4y)4 – x4
= [(3x + 4y)2]2 – (x2)2
= [(3x + 4y)2 + x2][(3x + 4y)2 – x2]
= [(3x + 4y)2 + x2][(3x + 4y) + x] [(3x + 4y) – x]
= {(3x + 4y)2 + x2} (3x + 4y + x)(3x + 4y – x)
= {(3x + 4y)2 + x2} (4x + 4y)(2x + 4y)
= {(3x + 4y)2 + x2} 4(x + y) 2 (x + 2y)
= 8{(3x + 4y)2 + x2} (x + y)(x + 2y)
Question: 28
Solve:
p2q2 – p4q4
Solution:
p2q2 – p4q4
= p2q2(1 – p2q2)
= p2q2 [1 – (pq)2]
= p2q2 (1 – pq)(1 + pq)
Question: 29
Solve:
3x3y – 243xy3
Solution:
3x3y – 243xy3
= 3xy(x2 – 81y2)
= 3xy[x2 – (9y)2]
= 3xy(x – 9y)(x + 9y)
Question: 30
Solve:
a4b4 – 16c4
Solution:
a4b4 – 16c4
= [(a2b2)2 – (4c2)2]
= (a2b2 + 4c2)(a2b2 – 4c2)
= (a2b2 + 4c2)[(ab)2 – (2c)2]
= (a2b2 + 4c2)(ab + 2c)(ab – 2c)
Question: 31
Solve:
x4 – 625
Solution:
x4 – 625
= (x2)2 – 252
= (x2 + 25)(x2 – 25)
= (x2+ 25)(x2 – 52)
= (x2 + 25)(x + 5)(x – 5)
Question: 32
Solve:
x4 – 1
Solution:
x4 – 1
= (x2)2 – 1
= (x2 + 1)(x2 – 1)
= (x2 + 1)(x + 1)(x – 1)
Question: 33
Solve:
49(a – b)2 – 25(a + b)2
Solution:
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)
= 2(a – 6b) 2(6a – b)
= 4(a – 6b)(6a – b)
Question: 34
Solve:
x – y – x2 + y2
Solution:
x – y – x2 + y2
= (x – y) + (y2 – x2)
= (x – y) + (y + x)(y – x)
= (x – y) – (y + x)(x – y) [since, (y – x) = – (x – y)]
= (x – y)[1 – (y + x)]
= (x – y)(1 – x – y)
Question: 35
Solve:
16(2x – 1)2 – 25y2
Solution:
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)
= (8x – 5y – 4)(8x + 5y – 4)
Question: 36
Solve:
4(xy + 1)2 – 9(x – 1)2
Solution:
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)
Question: 37
Solve:
(2x + 1)2 – 9x4
Solution:
(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(-x + 1) + 1(-x + 1)}(3x2 + 2x + 1)
= (-x + 1)(3x + 1)(3x2 + 2x + 1)
= – (x – 1)(3x + 1)(3x2 + 2x + 1)
Question: 38
Solve:
x4 – (2y – 3z)2
Solution:
x4 – (2y – 3z)2
= (x2)2 – (2y – 3z)2
= [x2 – (2y – 3z)][x2 + (2y – 3z)]
= (x2 – 2y + 3z)(x2 + 2y – 3z)
Question: 39
Solve:
a2 – b2 + a – b
Solution:
a2 – b2 + a –b = (a2 – b2) + (a – b)
= (a + b)(a – b) + (a – b)
= (a – b)(a + b + 1)
Question: 40
Solve:
16a4 – b4
Solution:
16a4 – b4
= (4a2)2 – (b2)2
= (4a2 + b2)(4a2 – b2)
= (4a2 + b2)[(2a)2 – b2]
= (4a2 + b2)(2a + b)(2a – b)
Question: 41
Solve:
a4 – 16(b – c)4
Solution:
a4 – 16(b – c)4
= (a2)2 – [4(b – c)2]2
= [a2 + 4(b – c)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)
Question: 42
Solve:
2a5 – 32a
Solution:
2a5 – 32a
= 2a(a4 – 16)
= 2a[(a2)2 – 42]
= 2a(a2 + 4)(a2 – 4)
= 2a(a2 + 4)(a2 – 22)
= 2a(a2 + 4)(a + 2)(a – 2)
= 2a(a – 2)(a + 2)(a2 + 4)
Question: 43
Solve:
a4b4 – 81c4
Solution:
a4b4 – 81c4
= (a2b2)2 – (9c2)2
= (a2b2 + 9c2)(a2b2 – 9c2)
= (a2b2 + 9c2)[(ab)2 – (3c)2]
= (a2b2+ 9c2)(ab + 3c)(ab – 3c)
Question: 44
Solve:
xy9 – yx9
Solution:
xy9 – yx9
= xy(y8 – x8)
= xy[(y4)2 – (x4)2]
= xy(y4 + x4)[(y2)2 – (x2)2]
= xy(y4 + x4)(y2 + x2)(y2 – x2)
= xy(y4 + x4)(y2 + x2)(y + x)(y – x)
Question: 45
Solve:
x3 – x
Solution:
x3 – x = x(x2 – 1)
= x(x – 1)(x + 1)
Question: 46
Solve:
18a2x2 – 32
Solution:
18a2x2 – 32
= 2(9a2x2 – 16)
= 2[(3ax)2 – 42]
= 2(3ax – 4)(3ax + 4)