Question 560674
The mistake occurred when you had x^3 = 1 and x^3 = 8 and concluded that the roots were 1 and 2. However x^3 = 1 has three roots of unity in the complex plane, namely


*[tex \LARGE x = e^0 = 1]


*[tex \LARGE x = e^{\frac{2\pi}{3}} = -\frac{1}{2} + \frac{\sqrt{3}}{2}i]


*[tex \LARGE x = e^{\frac{4\pi}{3}} = -\frac{1}{2} - \frac{\sqrt{3}}{2}i]


The roots for x^3 = 8 are similar, except twice as much (e.g. each root of x^3 = 1, multiplied by 2).