# Simplify an integer expression involving Ceiling and Root[] of third degree polynomial

I would like to simplify the following expression

    Ceiling[ToRadicals[Root[90 - 6 num + 65 #1 + 12 #1^2 + #1^3 &, 1]]]


with num an integer greater than or equal to 16. I tried using FullSimplify with the constraints above, but Mathematica couldn't find a simpler formula. The explicit Root is very complex (it involves third and sixth roots). However, I know there exist some identities involving the ceiling function (e.g. those here).

So, my question is: is there a (possibly simple) way to reduce the above in a form involving Log, Mod, Ceiling, Floor and similar functions (and possibly get rid of roots)? Any idea?

• with num an integer greater than (not equal to) 6 Ok, I tried: num = 7; Ceiling[ToRadicals[Root[90 - 6 num + 65 #1 + 12 #1^2 + #1^3 &, 1]]] and Mathematica said 0 Also tried with num=8 and got zero? Jan 16, 2017 at 23:16
• Sorry, my mistake: num >= 16. I fix Jan 16, 2017 at 23:19
• Ok, just tried 17 and tried 18 and get 1 ? !Mathematica graphics Jan 16, 2017 at 23:20
• Yes, right for both Jan 16, 2017 at 23:23
• @Piruzzolo What Nasser is trying to say is that, once a numerical value is assigned to num, Mathematica happily calculates the value of the expression quickly and with no need for FullSimplify etc. Are you looking for an explicit function of num? That seems difficult to obtain analytically, but easy to calculate on the spot. Perhaps you could expand on why you need this value; there may be other ways around the problem. Jan 17, 2017 at 1:25