Question

Factorize each of the following expressions:
$(a+b{)}^3-8(a-b{)}^3$

Answer 1

Given: $(a+b{)}^3-8(a-b{)}^3$

$\therefore (a+b{)}^3-8(a-b{)}^3$
$=(a+b{)}^3-[2(a-b){]}^3$
$=\left[\right(a+b)-2(a-b\left)\right]\left[\right(a+b{)}^2+(a+b)\times 2(a-b)+\left(2\right(a-b\left){)}^2\right]$

$[\therefore x^3-y^3=(x-y\left)\right(x^2+xy+y^2\left)\right]$

$=(a+b-2a+2b)[a^2+2ab+b^2+2a^2-2b^2+4(a^2-2ab+b^2\left)\right]$
$=(-a+3b)(7a^2-6ab+3b^2)$

Hence, $(a+b{)}^3-8(a-b{)}^3$
$=(-a+3b)(7a^2-6ab+3b^2).$