We only need to prove $P(1)=0$. Let $\omega_1$, $\omega_2$,$\omega_3$, and $\omega_4$ be four unit imaginary roots. Then, $x^4+x^3+x^2+x+1=\displaystyle\prod_{k=1}^{4}(x-\omega_k)$ We also have the following relationship: - $P(1) +\omega_1 Q(1) + \omega_1^2R(1) = 0$ - $P(1) +\omega_2 Q(1) + \omega_2^2R(1) = 0$ - $P(1) +\omega_3 Q(1) + \omega_3^2R(1) = 0$ This implies the equation $x^2 R(1) + x Q(1) + P(1) = 0 $ has three distinct imaginary roots: $\omega_1$, $\omega_2$, and $\omega_3$. This cannot be true unless $R(1)=Q(1)=P(1)=0$.