Mathematics
Factorise the following:
a3 - 3a2b + 3ab2 - 2b3
Factorisation
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Answer
a3 - 3a2b + 3ab2 - 2b3 = a3 - 3a2b + 3ab2 - b3 - b3.
We know that,
(a - b)3 = a3 - b3 - 3a2b + 3ab2.
∴ a3 - 3a2b + 3ab2 - b3 - b3 = (a - b)3 - b3.
We know that,
a3 - b3 = (a - b)(a2 + ab + b2).
∴ (a - b)3 - b3 = (a - b - b)[(a - b)2 + (a - b).b + b2]
= (a - 2b)(a2 + b2 - 2ab + ab - b2 + b2)
= (a - 2b)(a2 + b2 - ab).
Hence, a3 - 3a2b + 3ab2 - 2b3 = (a - 2b)(a2 + b2 - ab).
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