I've got a sequence of polynomials, and for each of them the roots obviously follow a definite pattern. Here are the roots of the 34th one
All others have their roots arranged in a similar trident-like shape.
The Magma calculator says that for all of them the Galois group is the full symmetric group (well, I've checked some at random, including this one).
Are there any general methods to deal with such cases? From the picture I would guess that there is (at least) some cyclic group of order 3 involved somehow, so I should be able to reduce the polynomial to some simpler pieces.
In case anybody needs it, here is the 34th polynomial itself:
1262647690+255700718588*x+13631224452890*x^2+308600267562954*x^3+3762852287730239*x^4+28505564353529723*x^5+147540954575690309*x^6+558174350534761902*x^7+1622640970960835388*x^8+3765746767401417227*x^9+7187405631689627039*x^10+11550206297273077580*x^11+15923705986345125919*x^12+19119250849681784936*x^13+20236485734134957625*x^14+19066890500802679414*x^15+16118028554700214562*x^16+12301011609949474371*x^17+8516914887762136145*x^18+5369486377372114741*x^19+3090436418891829689*x^20+1626416851942385301*x^21+783117607008746782*x^22+344842286920653399*x^23+138658742267629274*x^24+50768083784135219*x^25+16852561930994233*x^26+5039986223113741*x^27+1345771479765129*x^28+316733973856726*x^29+64474270662688*x^30+11025181048833*x^31+1508607262720*x^32+150323855360*x^33+8589934592*x^34
$$ 8589934592 x^{34}+150323855360 x^{33}+1508607262720 x^{32}+11025181048833 x^{31}+64474270662688 x^{30}+316733973856726 x^{29}+1345771479765129 x^{28}+5039986223113741 x^{27}+16852561930994233 x^{26}+50768083784135219 x^{25}+138658742267629274 x^{24}+344842286920653399 x^{23}+783117607008746782 x^{22}+1626416851942385301 x^{21}+3090436418891829689 x^{20}+5369486377372114741 x^{19}+8516914887762136145 x^{18}+12301011609949474371 x^{17}+16118028554700214562 x^{16}+19066890500802679414 x^{15}+20236485734134957625 x^{14}+19119250849681784936 x^{13}+15923705986345125919 x^{12}+11550206297273077580 x^{11}+7187405631689627039 x^{10}+3765746767401417227 x^9+1622640970960835388 x^8+558174350534761902 x^7+147540954575690309 x^6+28505564353529723 x^5+3762852287730239 x^4+308600267562954 x^3+13631224452890 x^2+255700718588 x+1262647690 $$