It is known that no two roots of the polynomial equation
$$\begin{aligned} \begin{aligned} \prod _{j=1}^n (x-r_j) + \prod _{j=1}^n (x+r_j) =0, \end{aligned} \end{aligned}$$
(1)
where
\(0 < r_1 \le r_2 \le \cdots \le r_n\), can be equal and the gaps between the roots of (
1) in the upper half-plane strictly increase as one proceeds upward, and for
\(0< h< r_k\), the roots of
$$\begin{aligned} (x-r_k-h)\prod _{\begin{array}{c} j=1\\ j\ne k \end{array}}^n(x-r_j) + (x+r_k+h)\prod _{\begin{array}{c} j=1\\ j\ne k \end{array} }^n (x+r_j) = 0 \end{aligned}$$
(2)
and the roots of (
1) in the upper half-plane lie alternatively on the imaginary axis. In this paper, we study how the roots and the critical points of (
1) and (
2) are located.