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\leκ\le 1\; ( if $N$ is even),\;\; -1\leκ\le N/(N-2)\; ( if $N$ is odd) hold. For the quadrinomial $q(z)=1+\kappa (z-z^{N-1})-z^N$, the corresponding inequalities are $$-N\mathrm{/}(N-2)\le \kappa \le 1(\text{ if }N\text{ is odd}),-N\mathrm{/}(N-2)\le \kappa \le N\mathrm{/}(N-2)(\text{ if }N\text{ is even})\mathrm{.}$$ In the cases of limiting values of the parameter $\kappa$, we provide factorization formulas for the corresponding quadrinomials. For example, when $N$ is odd and $\kappa =N\mathrm{/}(N-2)$, the following representation is valid: $$p(z)=(1+z{)}^3\prod _{j=1}^{(N-3)\mathrm{/}2}[1+z^2-2z{\gamma }_j],$$ where ${\gamma }_j=1-2{\nu }_j^2$ with $\{{\nu }_j{\}}_{j=1}^{(N-3)\mathrm{/}2}$ being the collection of positive roots of the equation $U_{N-2}^{\mathrm{\prime }}(x)=0$; here $$U_j(x)=U_j(\cos t)=\frac{\sin (j+1)t}{\sin t}=2^j{x}^j+\dots$$ are Chebyshev polynomials of the second kind and $U_j^{\mathrm{\prime }}(x)$ 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 Fejér polynomial, as well as construct certain univalent polynomials related to the polynomials $p(z)$ and $q(z)$.