Define $f(x) = x^6 + x^5 + 3x^4 +x^3 + 3x^2 + x + 1$. Find the largest prime factor of $f(19) + 1$ This problem is from a homework set of my class at source: Alphastar.academy. I believe there a number of ways to factor this and solve it, and I would appreciate it if I were able to see a couple methods on how to do this problem.
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2$\begingroup$ Have you tried any of those factoring methods? Remember that you should try to factorize $f(x)+1$ and not $f(x)$. $\endgroup$– Sarvesh Ravichandran IyerCommented Oct 23, 2020 at 6:18
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Conveniently, $f(x)+1$ factors as $$(x^2-x+1)(x^2+x+1)(x^2+x+2)$$ With $x=19$ this produces the three factors $343×381×382$, from which we work out that the largest prime factor is $191$ (of $382$).
answered Oct 23, 2020 at 6:21
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2$\begingroup$ No insight on how it was factorized though? $\endgroup$– MouradCommented Oct 23, 2020 at 7:50
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$\begingroup$ @Mourad I just plugged it into SymPy... it's three quadratics, so trial and error with "small" irreducible quadratic polynomials would suffice. $\endgroup$ Commented Oct 23, 2020 at 8:42