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Factorize:$a^3 - 3a^2b + 3ab^2 - b^3 + 8$
Given:
$a^3 - 3a^2b + 3ab^2 - b^3 + 8$
To do:
We have to factorize the given expression.
Solution:
We know that,
$a^3 + b^3 = (a + b) (a^2 - ab + b^2)$
$a^3 - b^3 = (a - b) (a^2 + ab + b^2)$
Therefore,
$a^3 - 3a^2b + 3ab^2 - b^3 + 8 = (a - b)^3 + (2)^3$
$= (a - b + 2) [(a -b)^2 - (a - b) \times 2 + (2)^2]$
$= (a- b + 2) (a^2 + b^2 -2ab - 2a + 2b + 4)$
Hence, $a^3 - 3a^2b + 3ab^2 - b^3 + 8 = (a- b + 2) (a^2 + b^2 -2ab - 2a + 2b + 4)$.
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