Question

Express the roots of the equation z3 −α3 = 0 in terms of α and w, where w is a complex cube root of unity. Use your answer to find the roots of the following equations in the form a + ib.

Answer 1

$$z^3-\alpha^3=0$$

$z^3-\alpha^3=(z-\alpha)(z^2+\alpha z+\alpha^2)$

Then

$$(z-\alpha)(z^2+\alpha z+\alpha^2)=0$$

z_1=\alpha\ or $z^2+\alpha z+\alpha^2=0$

$$z={-\alpha\pm\sqrt{\alpha^2-4\alpha^2}\over2}=\alpha({-1\pm j\sqrt{3}\over2})$$

Cube Root of Unity Value

$$w_1=1,\ real$$ $$w_2={-1- j\sqrt{3}\over2}, complex$$ $$w_3={-1+ j\sqrt{3}\over2}, \ complex$$

$$z_1=\alpha w_1=\alpha\cdot1=1\cdot\alpha+j\cdot0$$

$$z_2=\alpha w_2=\alpha\cdot{-1- j\sqrt{3}\over2}=-{1\over 2}\alpha-j\cdot{\alpha\sqrt{3}\over 2}$$

$$z_2=\alpha w_3=\alpha\cdot{-1+ j\sqrt{3}\over2}=-{1\over 2}\alpha+j\cdot{\alpha\sqrt{3}\over 2}$$