  Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic
(i.e., has only real roots) then $p+sp'$ is also hyperbolic for any $s\in
\mathbb{R}$. We study other perturbations of hyperbolic polynomials of the form
$p_a(z,s): =p(z) +\sum_{k=1}^d a_ks^kp^{(k)}(z)$. We give a full
characterization of those $a= (a_1, \dots, a_d) \in \mathbb{R}^d$ for which
$p_a(z,s)$ is a pencil of hyperbolic polynomials. We give also a full
characterization of those $a= (a_1, \dots, a_d) \in \mathbb{R}^d$ for which the
associated families $p_a(z,s)$ admit universal determinantal representations.
In fact we show that all these sequences come from special symmetric Toeplitz
matrices.
