How to reduce in sagemath a 40th degree equation to a 5th degree equation with a 6th degree equation? All three equations are mod 9

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Define a polynomial ring over the integers modulo nine.

Then define your polynomials.

They will reduce modulo nine automatically.

Then you can reduce one polynomial modulo the other one.

Step by step below.

Define the polynomial ring and the two polynomials:

sage: R.<x> = Zmod(9)['x']
sage: a = (22835963083295358096932575511191922182123945984*x^40
....:      + 456719261665907161938651510223838443642478919680*x^39
....:      - 949119715649463320903760169683914265694526504960*x^38
....:      - 74438103413079337596594904736638418838042150174720*x^37
....:      - 211966311223616472653064458230223650062515432325120*x^36
....:      + 5699423076536127631354075023203536650831985341628416*x^35
....:      + 28973596255008264172921747247784413655190430306795520*x^34
....:      - 271995774077019828088650443324511604201030917638062080*x^33
....:      - 1867697404327342199267901249360368498941431933516644352*x^32
....:      + 9056796651291403018168478696099136247382279833599868928*x^31
....:      + 79160624263907039138284570805006392617165984624088711168*x^30
....:      - 223551997508102686121059656283668963809997686306850734080*x^29
....:      - 2458991015411203904019832329162025635890321461058054651904*x^28
....:      + 4260693753881507670321728714982887826581938225381688475648*x^27
....:      + 59071293374791879685378907558855333817814530563406754217984*x^26
....:      - 65166805833928613316698470531587698330832804559521753071616*x^25
....:      - 1131757178880074916089566167507231334194665050735930352074752*x^24
....:      + 843273738240957848823393929862457096912314552519743678447616*x^23
....:      + 17612018891685450913773925408995240476326126279055993164791808*x^22
....:      - 9965070460907997689837650917987090430658787607861126978600960*x^21
....:      - 224803412228839542185624982039516671368355084084003095634247680*x^20
....:      + 115376363264002396798805411172610298752130761902895261355606016*x^19
....:      + 2360934383698977135600868106052207944902529771066700907860197376*x^18
....:      - 1308815542039402425703070866082728213909461400374939017803726848*x^17
....:      - 2033774651111365686863631947820879710270359595089530799861530624*x^16
....:      + 13558465130296871278633915980747022983007195978102160591142518784*x^15
....:      + 142298688018343424254395657862764431521447510737554686159576104960*x^14
....:      - 118833592326356595084649993445344253636179846197802558968828002304*x^13
....:      - 794065401834625006842542559429023479028263847219436978179087007744*x^12
....:      + 835819614910693852323042241683734160878949541950181684520436105216*x^11
....:      + 3426759016172348767255673212872820400903705074714452879983524184064*x^10
....:      - 4539247226522701303816038738989886579860962357232685201849757728768*x^9
....:      - 10838172582209345115431209424474859020815749759396823997114335887360*x^8
....:      + 18274547905763525031916419971549097309724446260463287119585027817472*x^7
....:      + 22538630042118410739462196918476504623707986472870448264236719144960*x^6
....:      - 51332294008917127137813530189611197158096252035341181816762615726080*x^5
....:      - 22103913759369896814721471064671155443726673363949038033115234369536*x^4
....:      + 89873238137891007192684349732347354457768316526153757296330427465728*x^3
....:      - 13267389606759431082806583489371907647806512881963825978855904528384*x^2
....:      - 73919625356285467948968028725559984622542638023960350194728622704640*x
....:      + 42792357763031228784543461920827458618601064112586804151160697595809)
sage: b = (262144*x^6 + 786432*x^5 - 8306688*x^4 - 17924096*x^3 + 96045792*x^2
....:      + 105138912*x - 405220671)

Check that they have been reduced modulo nine.

sage: a
7*x^40 + 5*x^39 + x^38 + 3*x^36 + 6*x^35 + 6*x^34 + 3*x^33
+ 8*x^31 + x^30 + 8*x^29 + 3*x^27 + 3*x^24 + x^22 + 5*x^21
+ 7*x^20 + 3*x^17 + 3*x^16 + 3*x^8 + 3*x^7 + 3*x^5 + 3*x^4
sage: b
x^6 + 3*x^5 + 6*x^4 + 7*x^3 + 6*x^2 + 3*x

Compute the remainder of one modulo the other.

sage: a % b
0