# Is Factor reliable to find factorization? [closed]

I have two polynomials given by:

pol1[m_]=-1841504878942953083744409324868187492994498167115966796875 +580928493770985107820326304616514809348845549291934394531250 m -75024502885706095121603114224530193128338549528835833460937500 m^2 +5298925657163453777955255232567622332487122280459454380645625000 m^3- 227798365109668201365365179085359713340345609333042534931445750000 m^4 + 6234643254347521026564392321121203254954895883725417799236105700000 m^5 -109931649216658523777148132034696040607135448091444662328102077240000 m^6 +1231078593841754871953530891323886882150825573827191207597910477904000 m^7 -8346690238810542695206381988003526167288154316253859581516221092819200 m^8 + 30361253286990267418750317340168323322357764935142152595016669455070720 m^9 -39216649815260853136388340128801670825683951373094135212510242806043648 m^10 -31200431178403164530283275139303069690890304131559778448027666206853120 m^11 -8219539462328877944794781176249479500300711570681926587860899836416000 m^12 – 1039886007634837239367140032604908347357863410399660015064806006005760 m^13 – 64775153967170825528688821497476316434508832977561783722518350807040 m^14 – 1601840981499800816473320846390578910974221468020505318875962441728 m^15;

and

pol2[n_]=-623131210119254319579340517520904541015625 -103073215809586842806311324238777160644531250000 n -5082539520065755531379068046808242797851562500000000 n^2 -69640546955654002405478386803833770751953125000000000000 n^3 +514010725273019292655872388921796286949218750000000000000000 n^4 +3468277793540502153144535767873711384125936700000000000000000000 n^5 – 32063473617674476689282475044826138831298084932723200000000000000000 n^6 + 82066923745429352350072890484827347893180065670261137408000000000000000 n^7 -89582869100748382783027823531551494377825489661793879565795328000000000000 n^8 +44345746669088315370881052440694611814402939222300825771895035002880000000000 n^9 -9167934205260936843174009374977136703014603122209490501821274650700677120000000 n^10 +448733774740221380068584217234641234848723966315423691985007171402365874995200000 n^11 +14088864426876823657094384891357446211331244875462673884479845968274373298094080000 n^12 + 115495643678639078009546442411048074550486137241422263776190261914146252810040115200 n^13 +347019796200710566141847612782344686029571076662005362793768187422495674076569272320 n^14 +282756961676188130021118727351278339021334858342543008344186700690079840295381368832 n^15;

If I try pol1[m]//Factor or pol2[n]//Factor I get the same expressions back without any factorization taking place. Does this guarantee that the two polynomials are irreducible, or could it happen that the large coefficients make the Factor routine break down? Should I look further into trying to factor them?

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Amazingly, it turns out that there is a built in Mathematica routine to check whether a polynomial is irreducible or not:

IrreduciblePolynomialQ[]

Applied to the above two polynomials it confirms that both are in fact irreducible.

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For univariate polynomials IrreducibleQ will simply use Factor`. For multivariate there are shortcuts available.
– Daniel Lichtblau
Sep 9 ’15 at 7:16