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. a) z^2-27=0

Answer 1

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

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

Then

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

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

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

Cube Root of Unity Value

$$w_1=1,\ real$$

$$w_2={-1- i\sqrt{3}\over2}, complex$$

$$w_3={-1+ i\sqrt{3}\over2}, \ complex$$

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

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

$$z_3=\alpha w_3=\alpha\cdot{-1+ i\sqrt{3}\over2}=-{1\over 2}\alpha+i\cdot{\alpha\sqrt{3}\over 2}$$