I need a few hints. Find the set W= {x belongs to Z (integers) such that
belongs to the integers}.
suppose that (x^{3} - 3x + 2)/(2x + 1) = k, where k is an integer.
then (8x^{3} - 24x + 16)/(2x + 1) = 8k. this is also an integer.
now 8x^{3} - 24x + 16 = (2x + 1)(4x^{2} - 2x - 11) + 27
so 8k = (8x^{3} - 24x + 16)/(2x + 1) = 4x^{2} - 2x - 11 + 27/(2x - 1).
since for ALL integers x, 4x^{2} - 2x - 11 is an integer, if 8k is to be an integer, we must have 2x - 1 is a divisor of 27, so:
2x - 1 = ±1, 3, 9 or 27.
this leads to W = {-14,-5,-2,-1,0,1,4,13} as emakarov conjectured.
This is where I always get stumped. I would have never guessed to multiply by 8. I just started really learning mathematics about two years ago, and I have a difficult time introducing numbers that help in solving the equation. Why 8 and not 3, 4 or some other number? Just curious. Thanks