Find roots of polynomial $$ p(x) = x^{20}+x^{19}+x^{18}+x^{17}+x^{16}+x^{15}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1 $$

The roots of polynomial $ p(x) $ are:

$$ \begin{aligned}x_1 &= 0.3653+0.9309i\\[1 em]x_2 &= 0.3653-0.9309i\\[1 em]x_3 &= -0.5+0.866i\\[1 em]x_4 &= -0.5-0.866i\\[1 em]x_5 &= -0.2225+0.9749i\\[1 em]x_6 &= -0.2225-0.9749i\\[1 em]x_7 &= 0.6235+0.7818i\\[1 em]x_8 &= 0.6235-0.7818i\\[1 em]x_9 &= 0.0747+0.9972i\\[1 em]x_{10} &= 0.0747-0.9972i\\[1 em]x_{11} &= -0.7331+0.6802i\\[1 em]x_{12} &= -0.7331-0.6802i\\[1 em]x_{13} &= -0.901+0.4339i\\[1 em]x_{14} &= -0.901-0.4339i\\[1 em]x_{15} &= 0.8262+0.5633i\\[1 em]x_{16} &= 0.8262-0.5633i\\[1 em]x_{17} &= 0.9556+0.2948i\\[1 em]x_{18} &= 0.9556-0.2948i\\[1 em]x_{19} &= -0.9888+0.149i\\[1 em]x_{20} &= -0.9888-0.149i \end{aligned} $$

Polynomial $ x^{20}+x^{19}+x^{18}+x^{17}+x^{16}+x^{15}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.

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