isi-entrance 2012 Q5

isi-entrance · India · solved Complex numbers 2 Roots of Unity and Cyclotomic Properties

Let $w$ be a primitive cube root of unity. Simplify $\dfrac{1}{z-3} + \dfrac{1}{z-3w} + \dfrac{1}{z-3w^2}$.

$$\frac{1}{z-3} + \frac{1}{z-3w} + \frac{1}{z-3w^2} = \frac{3z^2}{z^3 - 27}. \quad \text{(D)}$$

Let $w$ be a primitive cube root of unity. Simplify $\dfrac{1}{z-3} + \dfrac{1}{z-3w} + \dfrac{1}{z-3w^2}$.