Answer to Question #117244 in Algebra for Vincent zokah

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

Solution.

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

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

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

$z-\alpha=0$ or $z^2+\alpha z+\alpha^2=0;$

$z_1=\alpha;$ $D=\alpha^2-4\alpha^2=3\alpha^2;$

$z_2=\dfrac{-\alpha+\sqrt{3\alpha^2}}{2}=\alpha \dfrac{-1+i\sqrt{3}}{2};$

$z_3=\dfrac{-\alpha-\sqrt{3\alpha^2}}{2}=\alpha \dfrac{-1-i\sqrt{3}}{2};$

Cube Root of Unity Value:

$\omega_1=1$ - real;

$\omega_2= \dfrac{-1+i\sqrt{3}}{2}$ - complex;

$\omega_3= \dfrac{-1-i\sqrt{3}}{2}$ - complex;

$z_1=\alpha\sdot\omega_1=\alpha\sdot1=\alpha;$

$z_2=\alpha\sdot\omega_2=\alpha \dfrac{-1+i\sqrt{3}}{2}=-\dfrac{1}{2}\alpha+i\dfrac{\alpha\sqrt{3}}{2};$

$z_3=\alpha\sdot\omega_3=\alpha \dfrac{-1-i\sqrt{3}}{2}=-\dfrac{1}{2}\alpha-i\dfrac{\alpha\sqrt{3}}{2};$

Answer: $z_1=\alpha;$

$z_2=-\dfrac{1}{2}\alpha+i\dfrac{\alpha\sqrt{3}}{2};$

$z_3=-\dfrac{1}{2}\alpha-i\dfrac{\alpha\sqrt{3}}{2};$