Some properties of the quadrinomials $p(z)=1+\kappa(z+z^{N-1})+z^N$ and $q(z)=1+\kappa(z-z^{N-1})-z^N$

We show that all the zeros of the quadrinomial $p(z)=1+\kappa(z+z^{N-1})+z^N$ lie on the unit circle if and only if the inequalities $$-1\le\kappa\le\begin{cases}1,\quad\text{if }N\text{ is {\it even}},\\N/(N-2),\quad\text{if }N\text{ is {\it odd}}\end{cases}$$ hold. For the quadrinomial $q(z)=1+\kappa(z-z^{N-1})-z^N$, the corresponding inequalities~are $$-N/(N-2)\le\kappa\le\begin{cases}1,\quad\text{if } N \text{ is {\it odd}},\\N/(N-2),\quad\text{if }N \text{ is {\it even}}.\end{cases}$$ In the cases of limiting values of the parameter $\kappa$, we provide factorization formulas for the corresponding quadrinomials. For example, when $N$ is {\it odd} and $\kappa=N/(N-2)$, the following representation is valid: $p(z)=(1+z)^3\prod_{j=1}^{(N-3)/2}[1+z^2-2z\gamma_j]$, where $\gamma_j=1-2\nu_j^2$ with $\{\nu_j\}_{j=1}^{(N-3)/2}$ being the collection of positive roots of the equation $U'_{N-2}(x)=0$; here $U_j(x)=U_j(\cos t)=\frac{\sin(j+1)t}{\sin t}=2^j x^j+\cdots$ are Chebyshev polynomials of the second kind, and $U'_j(x)$ are their derivatives. Similar factorization formulas are also provided for $q(z)$. As an application of the obtained results, we give the factorization formulas for the derivative of the Fej\'er polynomial, as well as construct certain univalent polynomials related to the polynomials $p(z)$ and $q(z)$.