Factoring Variants of Chebyshev Polynomials with Minimal Polynomials of $\cos (\frac{2\pi }{d})$

We solve the problem of factoring polynomials $V_n(x)\pm 1$ and $W_n(x) \pm 1$where$V_n(x)$and$W_n(x)$ are Chebyshev polynomials of the third and fourth kinds. The method of proof is based on previous work by Wolfram [12] for factoring variants of Chebyshev polynomials of the first and second kinds, $T_n(x)\pm 1$ and $U_n(x)\pm 1$. We also show that, in general, there are no factorizations of variants of Chebyshev polynomials of the fifth and sixth kinds, $X_n(x)\pm 1$ and $Y_n(x)\pm 1$ using minimal polynomials of $\cos (\frac{2\pi }{d})$.