This paper applies Hilbert’s irreducibility theorem to the two-variable polynomial family f(x,t) = x^{2ℓ} - x^ℓ - t over Q. It is proved that f(x,t) is irreducible over Q(t), and that for almost all rational values t₀ ∈ Q, the specialized polynomial f(x,t₀) remains irreducible over Q. The family includes as a special case the golden-ratio polynomial P_ℓ(x) = x^{2ℓ} - x^ℓ - 1, whose splitting field is Q(φ^{1/ℓ}, ζ_ℓ), where φ = (1+√5)/2 is the golden ratio. The paper connects Hilbert’s theorem with continued-fraction approximants of φ, showing that for almost all rational convergents tᵢ = F_{i+1}/F_i, the specialization x^{2ℓ} - x^ℓ - tᵢ is irreducible over Q. Quantitative bounds are provided for the thin exceptional set and for convergence of powered continued-fraction sequences.