Factorize each of the following expressions:$(a - 2b)^3 - 512b^3$


Given:

$(a - 2b)^3 - 512b^3$

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,

$(a - 2b)^3 - 512b^3 = (a - 2b)^3 - (8b)^3$

$= (a - 2b - 8b) [(a - 2b)^2 + (a - 2b) \times 8b + (8b)^2]$

$= (a - 10b) [a^2 + 4b^2 - 4ab + 8ab - 16b^2 + 64b^2]$

$= (a - 10b) (a^2 + 4ab + 52b^2)$

Hence, $(a - 2b)^3 - 512b^3 = (a - 10b) (a^2 + 4ab + 52b^2)$.

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Updated on: 10-Oct-2022

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