We have \(\displaystyle P(x)= Q(x^{1111})\). Since \(\displaystyle Q(x) = 0\) for \(\displaystyle x_{k}=\cos\frac{2k\pi}{10}+i\sin\frac{2k\pi}{10}\), \(\displaystyle \qquad (k = 1, 2, 3, \dots, 9)\) and \(\displaystyle x_{k}^{1111}=\cos\frac{2222k\pi}{10}+i\sin\frac{2222k\pi}{10}=\cos(222k\pi+\frac{2k\pi}{10})+i\sin(222k\pi+\frac{2k\pi}{10})=x_{k}\), it follows that \(\displaystyle P(x^{k})=Q(x_{k}^{1111})=Q(x_{k})=0.\) Therefore, \(\displaystyle Q(x)|P(x)\). (Clapping)