ML Aggarwal Solutions Class 9 Math 4th Chapter Factorisation Exercise 4.5
ML Aggarwal Understanding ICSE Mathematics Class 9 Solutions Fourth Chapter Factorisation Exercise 4.5. APC Solution Class 9 Exercise 4.5.
(1) (i) 8x3+ y3
Solution:
8x3 + y3
= (2x)3+ y3
= (2x + y) (2x)2 – 2x.y + y2)
= (2x + y) (4x2 – 2xy + y2)
(ii) 64x3 – 125y3
Solution:
64x3 – 125y3
= (4x)3 – (5y)3
= (4x – 5y) ((4x)2 + 4x.5y + (5y)2)
= (4x – 5y) (16x2 + 20xy + 25y2)
(2) (i) 64x3 + 1
Solution:
64x3 + 1
= (4x)3+ 13
= (4x + 1) (4x)2 – 4x.1 + 12)
= (4x + 1) (16x2 – 4x + 1)
(ii) 7a3 + 56b3
Solution:
7a3 + 56b3
= 7 (a3 + 8b3)
= 7 (a3 + (2b)3)
= 7 ((a + 2b) (a2 – a.2b + (2b)2)
= 7 (a + 2b) (a2 – 2ab + 4b2)
(3) (i) x6/343 + 343/x6
Solution:
x6/343 + 343/x6
= (x2/7)3 + (7/x2)3
= (x2/7 + 7/x2) ((x2/7)2 – x3/7. 7/x3 + (7/x2)2)
= (x2/7 + 7/x2) (x4/4 + 44/x)
(ii) 8x3 – 1/27y3
Solution:
8x3 – 1/27y3
= (2x)3 – (1/3y)3
= (2x – 1/3y) (2x)2 + 2x.1/3y + (1/3y)2)
= (2x – 1/3y) (4x2 + 2x/3y + 1/9y2)
(4) (i) x2 + x5
Solution:
x2 + x5
= x2 (1 + x3)
= x2 (1 + x) (12 – 1.x + x2)
= x2 (1 + x) (1 – x + x2)
(ii) 32x4 – 500x
Solution:
32x4 – 500x
= 4x (8x3 – 125)
= 4x ((2x)3 – 53)
= 4x ((2x – 5) ((2x)2 + 2x.5 + 52))
= 4x (2x – 5) (4x2 + 10x + 25)
(5) (i) 27x3y3 – 8
Solution:
27x3y3 – 8
= (3xy)3 – 23
= (3xy – 2) ((3xy)2 + 3xy.2 + 22)
= (3xy – 2) (9x2y2 + 6xy + 4)
(ii) 27 (x + y)3 + 8 (2x – y)3
Solution:
27 (x + y)3 + 8 (2x – y)3
Let, x + y = a, 2x – y = b
∴ 27 (a)3 + 8b3
= (3a)3 + (2b)3
= (3a + 2b) (3a)2 – 3a.2b + (2b)2)
= (3a + 2b) (9a2 – 6ab + 4b2)
Putting the value of a & b we get,
= (3 (x + y) + 2 (2x + y)) (9 (x + y)2 – 6 (x + y) (2x – y) + 4 (2x – y)2}
= (3x + 3y + 4x – 2y) (9 (x2 + 2xy + y2) – 6 (2x2 – xy + 2xy – y2) + 4 (4x2 – 4xy + y2)
= (7x + y) (9x2 + 18xy + 9y2 – 12x2 – 6xy + 6y2 + 16x2 – 16xy + 4y2)
= (7x + y) (13x2 – 4xy 19y2)
(6) (i) a3 + b3 + a + b
Solution:
a3 + b3 + a + b
= (a + b) (a2 – ab + b2) + 1 (a + b)
= (a + b) (a2 – ab + b2 + 1)
(ii) a3 – b3 – a + b
Solution:
a3 – b3 – a + b
= (a – b) (a2 + ab + b2) – 1 (a – b)
= (a + b) (a2 + ab + b2 – 1)
(7) (i) x3 + x + 2
Solution:
x3 + x + 2
= x3 + 1 + x + 1
= (x + 1) (x2 – x + 1) + 1 (x + 1)
= (x + 1) (x2 – x + 1 + 1)
= (x + 1) (x2 – x + 2)
(8) (i) x3 + 6x2 + 12x + 16
Solution:
x3 + 6x2 + 12x + 16
= x3 + 6x2 + 12x + 8 + 8
= x3 + 3x2 × 2 + 3x.22 + 23 + 8
= (x + 2)3 + 8
= (x + 2)3 + 23
= (x + 2 + 2) ((x + 2)2 – (x + 2) 2 + 22)
= (x + 4) (x2 + 4 – 2x – 4 + 4)
= (x + 4) (x2 + 2x + 4)
(ii) a3 – 3a2b + 3ab2 – 2b3
Solution:
a3 – 3a2b + 3ab2 – 2b3
= a3 – 3a2b + 3ab2 – b3 – b3
= (a – b)3 – b3
= (a – b – b) (a3 + ab + b2)
= (a – 2b) (a2 + ab + b2)
(9) (i) 2a3 + 16b3 – 5a – 10b
Solution:
2a3 + 16b3 – 5a – 10b
= 2 (a3 + 8b3) – 5 (a + 2b)
= 2 (a3 + (2b)3) – 5 (a + 2b)
= 2 (a + 2b) (a2 – 2ab + 4b2) – 5 (a + b)
= 2 (a + 2b) (a2 – 2ab + 4b2 – 5)
(ii) a3 – 1/a3 – 2a + 2/a
Solution:
a3 – 1/a3 – 2a + 2/a
= a3 – 1/a3 – 2 (a – 1/a)
= (a – 1/a) (a2 + a .1/a + 1/a2) – 2 (a – 1/a)
= (a – 1/a) (a2 + 1 + 1/a2 – 2)
= (a – 1/a) (a2 + 1/a2 – 1)
(10) (i) a6 – b6
Solution:
a6 – b6
= (a3)2 – (b3)2
= (a3 + b3) (a3 – b3)
= (a + b) (a2 – ab + b2) (a – b) (a2 + ab + b2)
(ii) x6 – 1
Solution:
x6 – 1
= (x3)2 – 1
= (x3 + 1) (x3 – 1)
= (x + 1) (x2 – x + 1) (x – 1) (x2 + x + 1)
(11) (i) 64x6 – 729y6
Solution:
64x6 – 729y6
= (8x3)2 – (27y3)2
= (8x3 – 27y3) (8x3 + 27y3)
= ((2x)3 – (3y)3) (2x)3 + (3y)3)
= (2x – 3y) (4x2 + 6xy + 9y2) (2x + 3y) (4x2 – 6xy + 9y2)
(12) (i) 250 (a – b)3 + 2
Solution:
250 (a – b)3 + 2
= 2 (125 (a – b)3 + 1)
= 2 ((5 (a – b)) ((5 (a – b))2 + 5 (a – b).1 + 1)))
= 2 (5a – 5b) (25a2 – 5ab + 5a – 5b + 1)
= 2 (5a – 5b + 1) (25a2 – 5ab + 25b2 – 5ab + 1)
(ii) 32a2 x3 – 8b2x3 – 4a2 y3+ b2 y3
Solution:
32a2 x3 – 8b2x3 – 4a2 y3 + b2 y3
= 32a2 x3 – 4a2 y3 – 8b2 x3 + b2 y3
= 4a2 (8x3 – y3) – b2 (8x3 – y3)
= (4a2 – b2) ((2x)3 – y3)
= (2a + b) (2a – b) (2x – y) (4x2 + 2xy + y2)
(13) (i) x9 + y9
Solution:
x9 + y9
= (x3)3 + (y3)3
= (x3 + y3) (x6 – x3y3 + y6)
= (x + y) (x2 – 2y + y2) (x6 – x3y3 + y6)
(ii) x6 – 7x3 – 8
Solution:
x6 – 7x3 – 8
= x4 – 8x3 + x3 – 8
= x3 (x3 – 8) + 1 (x3 – 8)
= (x3 + 1) (x3 – 8)
= (x3 + 13) (x3 – 23)
= (x + 1) (x2 – x + 1) (x – 2) (x2 + 2x + 2)