Question 4
Let $0\lt \alpha\lt \pi$. Prove that there is a unique quadratic polynomial $x^2+ax+b$ that is a factor of
$$p_n(x)=\sin\alpha\,x^n-\sin(n\alpha)x+\sin(n\alpha-\alpha)$$
for every integer $n\gt 2$. Prove that no linear polynomial $x+c$ divides $p_n(x)$ for every $n\gt 2$.
$$p_n(x)=\sin\alpha\,x^n-\sin(n\alpha)x+\sin(n\alpha-\alpha)$$
for every integer $n\gt 2$. Prove that no linear polynomial $x+c$ divides $p_n(x)$ for every $n\gt 2$.