Let 𝖿 ∈ ℝ(𝑡)[𝑥] be given by 𝖿(𝑡, 𝑥) = 𝑥𝑛 + 𝑡 · g(𝑥) and β1 < ··· < β𝑚 the distinct real roots of the discriminant ∆(𝖿,𝑥)(𝑡) of 𝖿(𝑡, 𝑥) with respect to 𝑥. Let γ be the number of real roots of                    For any ξ > |βm|, if 𝑛−s is odd then the number of real roots of 𝖿(ξ,𝑥) is γ + 1, and if 𝑛−s is even then the number of real roots of 𝖿(ξ,𝑥) is γ, γ + 2 if ts > 0 or ts < 0 respectively. A special case of the above result is constructing a family of degree 𝑛 ≥ 3 irreducible polynomials over ℚ with many non-real roots and automorphism group S𝑛.