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Question
Factorize the following expressions:
(a - 2b)3 - 512b3
Solution
(a - 2b)3 - 512b3
= (a - 2b)3 - (8b)3
= (a - 2b - 8b)((a - 2b)2 + (a - 2b)8b + (8b)2) [ ∵ a3 - b3 = (a - b)(a2 + ab + b2)]
= (a -10b)(a2 + 4b2 - 4ab + 8b (a - 2b) + (8b)2 ) [ ∵ (a - b)2 = a2 + b2 - 2ab]
= (a -10b)(a2 + 4b2 - 4ab + 8ab -16b2 + 64b2)
= (a = 10b)(a2 + 68b2 -16b2 - 4ab + 8ab)
= (a -10b)(a2 + 52b2 + 4ab)
∴ (a - 2b)3 - 512b3 = (a -10b)(a2 + 4ab + 52b2)
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