# Thread: Find the Set W

1. ## Find the Set W

I need a few hints. Find the set W= {x belongs to Z (integers) such that

belongs to the integers}.

2. ## Re: Find the Set W

such that (x^3-3x+2)/(2x+1)

3. ## Re: Find the Set W

By testing numbers whose absolute value is < 10^6, it seems that W = {-14, -5, -2, -1, 0, 1, 4, 13}, but I am not sure how to prove it...

4. ## Re: Find the Set W

suppose that (x3 - 3x + 2)/(2x + 1) = k, where k is an integer.

then (8x3 - 24x + 16)/(2x + 1) = 8k. this is also an integer.

now 8x3 - 24x + 16 = (2x + 1)(4x2 - 2x - 11) + 27

so 8k = (8x3 - 24x + 16)/(2x + 1) = 4x2 - 2x - 11 + 27/(2x - 1).

since for ALL integers x, 4x2 - 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.

5. ## Re: Find the Set W

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

6. ## Re: Find the Set W

well first, i just did ordinary polynomial division over the rationals. then i took the least common denominator of all the coefficients i got to turn everything into integers. that least common denominator just happened to be 8.