Roots of a Gegenbauer polynomial

For $\alpha >0$, all $n$ roots of $C_n^{\alpha }$ are simple and lie in $(-1,1)$. If the product $q$ of its interior linear factors had degree below $n$, orthogonality would make $\int C_n^{\alpha }qw=0$, while $C_n^{\alpha }q=q^2(C_n^{\alpha }/q)$ has a constant strict sign away from finitely many points.