factor 512-(x+y)^3

Question

how do I factor 512-(x+y)^3

Answer 1

From the exponent around (x+y), it's leading me towards a difference of two cubes.

As it turns out 8^3 = 512. So

512 - (x+y)^3 = 8^3 - (x+y)^3

in general, the difference of two cubes factors into
(a-b)(a^2 + ab + b^2).

For our equation, this results in
[8 - (x+y)][8^2 + 8(x+y) + (x+y)^2] or
[8- (x+y)][64 + 8(x+y) + (x+y)^2]

if necessary, simplify further by distributing across (x+y) and expanding (x+y). This has been left for the student.