Factorize each of the following expressions:$a^3 + 3a^2b + 3ab^2 + b^3 - 8$

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,

$a^3 + 3a2^b + 3ab^2 + b^3 - 8 = (a + b)^3 - (2)^3$

$= (a + b - 2)[(a + b)^2 + (a+b)\times2 + (2)^2]$

$= (a + b-2) (a^2 + b^2 + 2ab + 2a + 2b + 4)$

$= (a + b - 2) [a^2 + b^2 + 2ab + 2(a + b) + 4]$

Hence, $a^3 + 3a2^b + 3ab^2 + b^3 - 8 = (a + b - 2) [a^2 + b^2 + 2ab + 2(a + b) + 4]$.

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Updated on: 10-Oct-2022

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