Chapter 7: Factorization Exercise – 7.6
Question: 1
Solve:
4x2 + 12xy + 9y2
Solution:
= (2x)2 + 2 x 2x x 3y + (3y)2
= (2x + 3y)2
= (2x + 3y) (2x + 3y)
Question: 2
Solve:
9a2 – 24ab + 16b2
Solution:
9a2 - 24ab + 16b2
= (3a)2 - 2 x 3a x 4b + (4b)2
= (3a - 4b)2
= (3a - 4b) (3a - 4b)
Question: 3
Solve:
p2q2 – 6qr + 9r2 = (pq)2 - 2 x pq x 3r + (3r)2
Solution:
p2q2 – 6qr + 9r2 = (pq)2 – 2 x pq x 3r + (3r)2
= (pq – 3r)2
= (pq – 3r) (pq – 3r)
Question: 4
Solve:
36a2 + 36a + 9
Solution:
36a2 + 36a + 9
= 9 (4a2 + 4a + 1) = 9{(2a)2 + 2 x 2a x 1 + 12}
= 9 (2a + 1)2
= 9 (2a + 1) (2a + 1)
Question: 5
Solve:
a2 + 2ab + b2 - 16
Solution:
a2 + 2ab + b2 - 16
= a2 + 2 x a x b + b2 – 16
= (a + b)2 – 42
= (a + b – 4) (a + b + 4)
Question: 6
Solve:
9z2 - x2 + 4xy – 4y2
Solution:
9z2 – x2 + 4xy – 4y2
= 9z2 - (x2 – 4xy + 4y2)
= 9z2 - [x2 – 2x × 2y + (2y)2]
= (3z)2- (x - 2y)2
= [3z - (x - 2y)] [3z + (x - 2y)]
= (3z - x + 2y) (3x + x -2y)
= (x - 2y + 3z) (-x + 2y + 3z)
Question: 7
Solve:
9a4 - 24a2b2 + 16b4 - 256
Solution:
9a4 - 24a2b2 + 16b4- 256
= (9a4 - 24a2b2 + 16b4) - 256
= [(3a2)2- 2 x 3a2 x 4b2 + (4b2)2] -162
= (3a2 - 402)2 -162
= [(3a2 – 4b2) -16] [(3a2 - 42) + 16]
= (3a2 - 4b2 -16) (3a2 - 4b2 + 16)
Question: 8
Solve:
16 - a6 + 4a3b3 – 4b6
Solution:
16 – a6 + 4a3b3 – 4b6
= 16 – (a6 – 4a3b3 + 4b6)
= 42 – [(a3)2 – 2 x a3 x 2b3 + (2b3)2]
= 42 – (a3 – 2b3)2
= [4 – (a3 – 2b3)] [4 + (a3 – 2b3)]
= (4 – a3 – 2b3) (4 + a3 – 2b3)
= (a3 – 2b3 + 4) (– a3 – 2b3 + 4)
Question: 9
Solve:
a2 – 2ab + b2 – c2
Solution:
a2 – 2ab + b2 – c2
= (a2 – 2ab + b2) – c2
= (a2 – 2 x a x b + b2) – c2
= (a – b)2 – c2
= [(a – b) – c][ (a – b) + c]
= (a – b – c) (a – b + c)
Question: 10
Solve:
x2 + 2x + 1- 9y2
Solution:
x2 + 2x + 1 – 9y2
= (x2 + 2x + 1) – 9y2
= (x2 + 2× x x 1 + 1) – 9y2
= (x + 1)2 – (3y)2
= [(x + 1) – 3y] [(x + 1) – 3y]
= (x + 1 – 3y) (x + 1 + 3y)
= (x + 3y + 1) (x – 3y + 1)
Question: 11
Solve:
a2 + 4ab + 3b2
Solution:
a2 + 4ab + 3b2
= a2 + 4ab + 4b2 – b2
= [a2 + 2 x a x 2b + (2b)2] – b2
= (a + 2b)2 – b2
= [(a + 2b) – b] [(a + 2b) + b]
= (a + 2b – b)(a + 2b + b)
= (a + b) (a + 3b)
Question: 12
Solve:
96 – 4x – x2
Solution:
96 - 4x – x2
= 100 - 4 – 4x – x2
= 100 - (x2 + 4x + 4)
= 100 - (x2 + 2 x x x 2 + 22)
= 102 – (x + 2)2
= [10 – (x + 2)] [10 + (x + 2)]
= (10 – x – 2)(10 + x + 2)
= (8 – x) (12 + x)
= (x + 12) (-x + 8)
Question: 13
Solve:
a4 + 3a2 + 4
Solution:
a4 + 3a2 + 4
= a4 + 4a2 – a2 + 4
= (a4 + 4a2 + 4) – a2
= [(a2)2 + 2 x a2 x 2 + 22] – a2
= (a2 + 2)2 – a2
= [(a2 + 2) – a][(a2 + 2) + a]
= (a2 – a + 2)(a2 + a + 2)
Question: 14
Solve:
4x4 + 1
Solution:
4x4 + 1
= 4x4 + 4x2 + 1 – 4x2
= [(2x2)2 + 2 x 2x2 x 1 + 1] – 4x2
= (2x2 + 1)2 – (2x)2
= [(2x2 + 1) – 2x] [(2x2 + 1) + 2x]
= (2x2 – 2x + 1)( 2x2 + 2x + 1)
Question: 15
Solve:
4x4 + y4
Solution:
4x4 + y4
= 4x4 + 4x2 + y4 – 4x2y2
= [(2x2)2 + 2 x 2x2 x y + (y2) 2] – (2xy)2
= (2x2 + y2)2 – (2xy)2
= [(2x2 + y2) – 2xy] [(2x2 + y2) + 2xy]
= (2x2 – 2xy + y2)( 2x2 + 2xy + y2)
Question: 16
Solve:
(x + 2)2 – 6(x + 2) + 9
Solution:
(x + 2)2 – 6(x + 2) + 9
= (x + 2)2 – 2 x (x + 2) x 3 + 32
= [(x + 2) – 3]2
= (x + 2 – 3)2
= (x – 1)2
= (x – 1)(x – 1)
Question: 17
Solve:
25 – p2 – q2 – 2pq
Solution:
25 – p2 – q2 – 2pq
= 25 – (p2 + 2pq + q2)
= 52 – (p2 + 2 x p x q + q2)
= 52 – (p + q)2
= [5 – (p + q)] [5 + (p + q)]
= (5 – p + q) (5 + p + q)
= - (p + q - 5)(p + q + 5)
Question: 18
Solve:
x2 + 9y2 – 6xy – 25a2
Solution:
x2 + 9y2 – 6xy – 25a2
=(x2 – 6xy + 9y2) – 25a2
= [x2 – 2 x x x 3y + (3y)2] – 25a2
= (x – 3y)2 – (5a)2
= [(x – 3y) – 5a][(x -3y) + 5a]
= (x – 3y – 5a)( x – 3y + 5a)
Question: 19
49 – a2 + 8ab – 16b2
Solution:
49 – a2 + 8ab – 16b2
= 49 – (a2 – 8ab + 16b2)
= 49 – [a2 – 2 x a x 4b + (4b2)]
= 72 – (a – 4b2)
= [7 – (a – 4b)][7 + (a – 4b)]
= (7 – a + 4b)( 7 + a – 4b)
= – (a – 4b – 7)(a – 4b + 7)
= – (a – 4b + 7)(a – 4b – 7)
Question: 20
Solve:
a2 – 8ab + 16b2 – 25c2
Solution:
a2 – 8ab + 16b2 – 25c2
= (a2 – 8ab + 16b2) – 25c2
= [a2 – 2 x a x 4b + (4b)2] – 25c2
= (a – 4b)2 – (5c)2
= [(a – 4b) – 5c] [(a – 4b)2 + 5c]
= (a – 4b – 5c) (a – 4b + 5c)
Question: 21
Solve:
x2 – y2 + 6y – 9
Solution:
x2 – y2 + 6y – 9
= x2 – (y2 + 6y – 9)
= x2 – (y2 – 2 x y x 3 + 32)
= x2 – (y – 3)2
= [x – (y – 3)] [x + (y – 3)]
= (x – y + 3)(x + y - 3)
Question: 22
Solve:
25x2 – 10x + 1 – 36y2
Solution:
25x2 – 10x + 1 – 36y2
= (25x2 – 10x + 1) – 36y2
= [(5x)2 – 2 x 5x x 1 + 1] – 36y2
= (5x – 1)2 – (6y)2
= [(5x – 1) – 6y] [(5x – 1) + 6y]
= (5x – 1 – 6y)( 5x – 1 + 6y)
= (5x – 6y – 1)( 5x + 6y – 1)
Question: 23
Solve:
a2 – b2 + 2bc – c2
Solution:
a2 – b2 + 2bc – c2
= a2 – (b2 – 2bc + c2)
= a2 – (b2 – 2 x b x c + c2)
= a2 – (b – c)2
= [a – (b – c)][ a + (b – c)]
= (a – b + c)(a + b – c)
Question: 24
Solve:
a2 + 2ab + b2 – c2
Solution:
a2 + 2ab + b2 – c2
= (a2 + 2ab + b2) – c2
= (a2 + 2 x a x b + b2) – c2
= (a + b)2 – c2
= [(a + b) – c] [(a + b) + c]
= (a + b – c) (a + b + c)
Question: 25
Solve:
49 – x2 – y2 + 2xy
Solution:
49 – x2 – y2 + 2xy
= 49 – (x2 + 2xy – y2)
= 72 – (x – y)2
= [7 – (x – y)] [7 + (x – y)]
= (7 – x + y)(7 + x – y)
= (x – y + 7)(y – x + 7)
Question: 26
Solve:
a2 + 4b2 – 4ab – 4c2
Solution:
a2 + 4b2 – 4ab – 4c2
= (a2 + 4b2 – 4ab) – 4c2
= [a2 – 2 x a x 2b + (2b)2] – 4c2
= (a – 2b)2 – (2c)2
= [(a – 2b) – 2c] [(a – 2b) + 2c]
= (a – 2b – 2c)(a – 2b + 2c)
Question: 27
Solve:
x2 – y2 – 4xz + 4z2
Solution:
x2 – y2 – 4xz + 4z2
= (x2 – 4xz + 4z2) – y2
= (x – 2z)2 – y2
= [(x – 2z) – y] [(x – 2z) + y]
= (x – 2z – y)(x – 2z + y)
= (x + y – 2z)(x – y – 2z)
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