We show explicit forms of the Bertini-Noether reduction theorem and of the Hilbert irreducibility theorem. Our approach recasts in a polynomial context the geometric Grothendieck good reduction criterion and the congruence approach to HIT for covers of the line. A notion of "bad primes" of a polynomial P ∈ ℚ[T,Y] irreducible over ℚ̅ is introduced, which plays a central and unifying role. For such a polynomial P, we deduce a new bound for the least integer t₀ ≥ 0 such that P(t₀,Y) is irreducible in ℚ[Y]: in the generic case for which the Galois group of P over ℚ̅(T) is Sₙ ($n=deg_Y(P)$), this bound only depends on the degree of P and the number of bad primes. Similar issues are addressed for algebraic families of polynomials $P(x₁,...,x_s,T,Y)$.