Find roots of polynomial $$ p(x) = x^{10}+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0.4154+0.9096i\\[1 em]x_2 &= 0.4154-0.9096i\\[1 em]x_3 &= -0.6549+0.7558i\\[1 em]x_4 &= -0.6549-0.7558i\\[1 em]x_5 &= -0.1423+0.9898i\\[1 em]x_6 &= -0.1423-0.9898i\\[1 em]x_7 &= 0.8413+0.5406i\\[1 em]x_8 &= 0.8413-0.5406i\\[1 em]x_9 &= -0.9595+0.2817i\\[1 em]x_{10} &= -0.9595-0.2817i \end{aligned} $$Explanation
Polynomial $ 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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