Find roots of polynomial $$ p(x) = x^{10}-10x^9+40x^8-80x^7+80x^6-32x^5-1 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -0.4142\\[1 em]x_2 &= 2.4142\\[1 em]x_3 &= 0.364+0.4621i\\[1 em]x_4 &= 0.364-0.4621i\\[1 em]x_5 &= -0.2098+0.3931i\\[1 em]x_6 &= -0.2098-0.3931i\\[1 em]x_7 &= 1.636+0.4621i\\[1 em]x_8 &= 1.636-0.4621i\\[1 em]x_9 &= 2.2098+0.3931i\\[1 em]x_{10} &= 2.2098-0.3931i \end{aligned} $$Explanation
Polynomial $ x^{10}-10x^9+40x^8-80x^7+80x^6-32x^5-1 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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