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Question
Factorize the following expressions:
(a + b)3 – 8(a – b)3
Solution
(a + b)3 - 8(a - b)3
= (a + b)3 - [2(a - b)]3
= (a + b)3 - (2a - 2b)3 [Using a3 - b3 = (a - b)(a2 + ab + b2 ) ]
= (a + b - (2a - 2b))((a + b)2 + (a + b)(2a - 2b) + (2a - 2b)2)
= (a + b - 2a + 2b)(a2 + b2 + 2ab + (a + b)(2a - 2b) + (2a - 2b)2) [∵ (a + b)2 = a2 + b2 + 2ab]
= (3b - a)(a2 + b2 + 2ab + 2a2 - 2ab + 2ab - 2b2 + (2a - 2b)2)
= (3b - a)(3a2 + 2ab - b2 + (2a - 2b)2 )
= (3b - a)(3a2 + 2ab - b2 + 4a2 + 4b2 - 8ab) [∵ (a - b)2 = a2 + b2 - 2ab]
= (3b - a )(3a2 + 4a2 - b2 + 4b2 + 2ab - 8ab)
= (3b - a )(7a2 + 3b2 - 6ab)
∴ (a + b)3 - 8(a - b)3 = (-a + 3b)(7a2 - 6ab + 3b2)
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