Decide whether some given polynomials are irreducible over QQ

Decide whether the following polynomials are irreducible in \mathbb{Q}[x].

  1. a(x) = x^3 + 2x^2 + 8x + 2
  2. b(x) = x^3 + 2x^2 + 2x + 4
  3. c(x) = x^3 + x^2 + x + 1
  4. d(x) = x^3 + 14
  5. e(x) = 5x^9 - 41
  6. f(x) = x^2 + 5x + 25
  7. g(x) = 5x^5 + 30x^4 + 42x^3 + 6x + 12

  1. a(x) is Eisenstein at 2 and thus is irreducible.
  2. b(x) = (x^2+2)(x+2) is reducible.
  3. c(x) = (x^2+1)(x+1) is reducible.
  4. d(x) is Eisenstein at 7 and thus is irreducible.
  5. e(x) is Eisenstein at 41 and thus is irreducible.
  6. f(x-1) = x^2 + 3x + 21 is Eisenstein at 3 and thus is irreducible. So f(x) is also irreducible.
  7. g(x) is Eisenstein at 3 and thus is irreducible.
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