lean_workbook_plus_76831

Note that $x - \omega$ and $x - \omega^2$ are the roots of $P_1(x)=x^2 + x + 1=0$ at $x= \omega, \omega^2$ , where $\omega^3 = 1$ . So those 2 linear polynomials are the factors of $P_1$ . Also, $\omega \neq 1$ and the $x - \omega$ and $x - \omega^2$ also divide $P_2(x)=x^7 + x^2 + 1$ . Thus, all the factors of $P_1$ are also factors of $P_2$ , so $P_1$ itself must divide $P_2$ .