Factorize the following expressions: (a-2 b)^3-512 b^3 - Vidyadip

Question

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

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Answer

$(a-2 b)^3-512 b^3$
$=(a-2 b)^3-(8 b)^3$
$=(a-2 b-8 b)\left((a-2 b)^2+(a-2 b) 8 b+(8 b)^2\right)$
$\therefore\left[a^3-b^3=(a-b)\left(a^2+a b+b^2\right)\right]$
$=(a-10 b)\left(a^2+4 b^2-4 a b+8 a b-16 b^2+64 b^2\right)$
$=(a-10 b)\left(a^2+52 b^2+4 a b\right)$
$\therefore(a-2 b)^3-512 b^3=(a-10 b)\left(a^2+52 b^2+4 a b\right)$

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