Question

A polynomial P is given.
P(x) = x6 − 1
(a) Find all zeros of P, real and complex. (Enter your answers as a comma-separated list. If a root has multiplicity greater than one, only enter the root once.)

Answer 1

x^1=(x-1)(x+1)(x^2-x+1)(x^2+x+1)
real roots are -1,1,...

Answer 2

\[x^6-1=0\]\[x^6=1=e^{i2\pi k}\quad,\quad k=0,1,2,3,4,5\]\[x_{k+1}=e^{i2\pi k/6}=e^{i\pi k/3}\quad,\quad k=0,1,2,3,4,5\]\[x_1=e^{i0}=1\]\[x_2=e^{i\pi /3}=\frac{1}{2}+i\frac{\sqrt{3}}{2}\]\[x_3=e^{i2\pi /3}=-\frac{1}{2}+i\frac{\sqrt{3}}{2}\]\[x_4=e^{i\pi }=-1\]\[x_5=e^{i4\pi /3}=-\frac{1}{2}-i\frac{\sqrt{3}}{2}\]\[x_6=e^{i5\pi /3}=\frac{1}{2}-i\frac{\sqrt{3}}{2}\]