Factorise each of the following:
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 - 1/216 - 9/2 p2 + 1/4 p
Answer
(i) 8a3 + b3 + 12a2b + 6ab2
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) 8a3 - b3 - 12a2b + 6ab2
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) 27 - 125a3 - 135a + 225a2
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) 64a3 - 27b3 - 144a2b + 108ab2
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) 27p3 - 1/216 - 9/2 p2 + 1/4 p
Using identity, (a - b)3 = a3 - b3 - 3a2b + 3ab2
27p3 - 1/216 - 9/2 p2 + 1/4 p
= (3p)3 - (1/6)3 - 3(3p)2(1/6) + 3(3p)(1/6)2
= (3p - 1/6)3
= (3p - 1/6)(3p - 1/6)(3p - 1/6)
