Question

The factors for Choose one of the factors of 27x3 + 512y3 are ...
(3x + 8y) and (9x^2 - 24xy + 64x^2) ?

Answer 1

27x^3 +512y^3 = 0
27x^3=-512y^3
-y/x =(27/512)^(1/3)
y/x =(-3/8)
3x+8y=0
Therefore, it is a factor of 27x^3 +512y^3

use long division,

(27x^3 +512y^3)/(3x+8y)

the quotient is not (9x^2 - 24xy + 64x^2) but (9x^2 - 24xy + 64y^2)

Answer 2

27x3 + 512y3
= (3x)^3 + (8y)^3
= (3x+8y) [ (3x)^2 - (3x)(8y) + (8y)^2 ]
= (3x+8y) (9x^2 -24xy + 64y^2)

Identity used: \(a^3+b^3 = (a+b)(a^2-ab+b^2)\)