The roots of the cubic equation (z + ab)^3 = | vedclass

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(B) Taking cube roots on both sides of the equation $(z + ab)^3 = a^3$:$z + ab = a, a\omega, a\omega^2$,where $\omega$ is the cube root of unity.Thus,

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$\sqrt{3} |a|$

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(B) Taking cube roots on both sides of the equation $(z + ab)^3 = a^3$:$z + ab = a, a\omega, a\omega^2$,where $\omega$ is the cube root of unity.Thus,the roots are $z_1 = a - ab$,$z_2 = a\omega - ab$,and $z_3 = a\omega^2 - ab$.The side length of the triangle is the distance between any two roots,for example,$|z_1 - z_2|$.$|z_1 - z_2| = |(a - ab) - (a\omega - ab)| = |a(1 - \omega)|$.Since $1 - \omega = 1 - (-\frac{1}{2} + i\frac{\sqrt{3}}{2}) = \frac{3}{2} - i\frac{\sqrt{3}}{2}$.$|1 - \omega| = \sqrt{(\frac{3}{2})^2 + (-\frac{\sqrt{3}}{2})^2} = \sqrt{\frac{9}{4} + \frac{3}{4}} = \sqrt{\frac{12}{4}} = \sqrt{3}$.Therefore,the side length is $|a| \cdot \sqrt{3} = \sqrt{3} |a|$.

The roots of the cubic equation $(z + ab)^3 = a^3$,where $a \neq 0$,represent the vertices of a triangle of sides of length:

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