Question
Factorize the following expressions:
$(a + b)^3 - 8(a - b)^3$
$(a + b)^3 - 8(a - b)^3$
✓
Answer
$= (a + b)^3 - [2(a - b)]^3 = (a + b)^3 - [2a - 2b]^3 $
$= (a + b - (2a - 2b))((a + b)^2 + (a + b)(2a - 2b) + (2a - 2b)^2)$
$\therefore$ $[a^3 - b^3 = (a - b)(a^2 + ab + b^2)]$
$= (a + b - 2a + 2b)(a^2 + b^2 + 2ab + (a + b)(2a - 2b) + (2a - 2b)^2) $
$=(a + b - 2a + 2b)(a^2 + b^2 + 2ab + 2a^2 - 2ab + 2ab - 2b^2 + (2a - 2b)^2) $
$= (3b - a)(3a^2 + 2ab - b^2 + (2a - 2b)^2) $
$= (3b - a)(3a^2 + 2ab - b^2 + 4a^2 + 4b^2 - 8ab) $
$= (3b - a)(3a^2 + 4a^2 - b^2 + 4b^2 - 8ab + 2ab) $
$= (3b - a)(7a^2 +3b^2- 6ab)$
$\therefore$ $(a + b)^3 - 8(a - b)^3 = (3b - a)(7a^2 + 3b^2 - 6ab)$
$= (a + b - (2a - 2b))((a + b)^2 + (a + b)(2a - 2b) + (2a - 2b)^2)$
$\therefore$ $[a^3 - b^3 = (a - b)(a^2 + ab + b^2)]$
$= (a + b - 2a + 2b)(a^2 + b^2 + 2ab + (a + b)(2a - 2b) + (2a - 2b)^2) $
$=(a + b - 2a + 2b)(a^2 + b^2 + 2ab + 2a^2 - 2ab + 2ab - 2b^2 + (2a - 2b)^2) $
$= (3b - a)(3a^2 + 2ab - b^2 + (2a - 2b)^2) $
$= (3b - a)(3a^2 + 2ab - b^2 + 4a^2 + 4b^2 - 8ab) $
$= (3b - a)(3a^2 + 4a^2 - b^2 + 4b^2 - 8ab + 2ab) $
$= (3b - a)(7a^2 +3b^2- 6ab)$
$\therefore$ $(a + b)^3 - 8(a - b)^3 = (3b - a)(7a^2 + 3b^2 - 6ab)$
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