Question
Factorize the following expressions:
$(x + 2)^3 + (x - 2)^3$
$(x + 2)^3 + (x - 2)^3$
✓
Answer
$= (x + 2 + x - 2)((x + 2)^2- (x + 2)(x - 2) + (x - 2)^2)$
$\big[\therefore$ [$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$\big]$
$= 2x(x^2 + 4x + 4 - (x + 2)(x - 2) + x^2 - 4x + 4)$
$= 2x(2x^2 + 8 - (x^2 - 2^2))$
$\big[\therefore$ $(a + b)(a - b) = a^2 - b^2$$\big]$
$= 2x(2x^2 + 8 - x^2 + 4)$
$= 2x(x^2 + 12)$
$\therefore$ $(x + 2)^3 + (x - 2)^3 = 2x(x^2 + 12)$
$\big[\therefore$ [$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$\big]$
$= 2x(x^2 + 4x + 4 - (x + 2)(x - 2) + x^2 - 4x + 4)$
$= 2x(2x^2 + 8 - (x^2 - 2^2))$
$\big[\therefore$ $(a + b)(a - b) = a^2 - b^2$$\big]$
$= 2x(2x^2 + 8 - x^2 + 4)$
$= 2x(x^2 + 12)$
$\therefore$ $(x + 2)^3 + (x - 2)^3 = 2x(x^2 + 12)$
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