허근 ω 교대합

[출제의도] 삼차방정식 이해하기 $x ^{3} = 1$의 한 허근을 $\omega$라 하면 $\omega ^{3} = 1$, $\omega ^{2} + \omega + 1 = 0$이다. $\displaystyle f ( 1 ) = \frac{\omega}{1 + \omega ^{2}} = \frac{\omega}{- \omega} = - 1 = f ( 4 ) = f ( 7 ) = f ( 10 ) = f ( 13 )$ $\displaystyle f ( 2 ) = \frac{\omega ^{2}}{1 + \omega ^{4}} = \frac{\omega ^{2}}{1 + \omega} = - 1 = f ( 5 ) = f ( 8 ) = f ( 11 )$ $\displaystyle f ( 3 ) = \frac{\omega ^{3}}{1 + \omega ^{6}} = \frac{1}{1 + 1} = \frac{1}{2} = f ( 6 ) = f ( 9 ) = f ( 12 )$ $\therefore f ( 1 ) - f ( 2 ) + \cdots - f ( 6 ) = 0$ $\therefore \text{준식} = f ( 13 ) = - 1$