Q.8. Factorise each of the following:

(i) 8a3 + b3 + 12a2b + 6ab2

(ii) 8a3 - b3 - 12a2b + 6ab2

(iii) 27 - 125a3 - 135a + 225a2

(iv) 64a3 - 27b3 - 144a2b + 108ab2

v 27p3 -1216-92p2 + 14p

Solution:

(i) Using identity, (a + b)3 = a3 + b3 + 3a2b + 3ab2

8a3 + b3 + 12a2b + 6ab2

= (2a)3 + b3 + 3(2a)2b + 3(2a)(b)2

= (2a + b)3

= (2a + b)(2a + b)(2a + b)

(ii) Using identity, (a - b)3 = a3 - b3 - 3a2b + 3ab2

8a3 - b3 - 12a2b + 6ab2

= (2a)3 - b3 - 3(2a)2b + 3(2a)(b)2

= (2a - b)3

= (2a - b)(2a - b)(2a - b)

(iii) Using identity, (a - b)3 = a3 - b3 - 3a2b + 3ab2

27 - 125a3 - 135a + 225a2

= 33 - (5a)3 - 3(3)2(5a) + 3(3)(5a)2

= (3 - 5a)3

= (3 - 5a)(3 - 5a)(3 - 5a)

(iv) Using identity, (a - b)3 = a3 - b3 - 3a2b + 3ab2

64a3 - 27b3 - 144a2b + 108ab2

= (4a)3 - (3b)3 - 3(4a)2(3b) + 3(4a)(3b)2

= (4a - 3b)3

= (4a - 3b)(4a - 3b)(4a - 3b)

(v) Using identity, (a - b)3 = a3 - b3 - 3a2b + 3ab2

27p3 -1216-92p2 + 14p

= 3p3 - 163 - 33p216+ 33p162

= 3p -163

= 3p -163p -163p -16