I recently found a problem saying "Find 2 irrational numbers such that their sum and product is both rational."
After a while I noticed any pair like $(a+\sqrt{b},a-\sqrt{b})$ work .From this I could easily say this statement is true for any even integer.But then I thought whether the same is true for odd integers. I realized proving it for $3$ would imply the statement being true for all odd integers. I tried to work with similar expression as above.I also tried to work with the cubic polynomial.But I couldn't make any significant progress.
You could take $\{j\:\sqrt[2k+1]{2}|1\le j\le 2k\}\cup\{-k(2k+1)\:\sqrt[2k+1]{2}\}$. The sum is $0$; the product is $-(2k+1)!2k$.