Find roots of polynomial $$ p(x) = 5x^7-1+2x-8x^{13}+20x^3 $$

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

$$ \begin{aligned}x_1 &= 0.28\\[1 em]x_2 &= 1.1385\\[1 em]x_3 &= -1.1442\\[1 em]x_4 &= -0.1388+0.3985i\\[1 em]x_5 &= -0.1388-0.3985i\\[1 em]x_6 &= -0.3807+1.0324i\\[1 em]x_7 &= -0.3807-1.0324i\\[1 em]x_8 &= 0.8537+0.6421i\\[1 em]x_9 &= 0.8537-0.6421i\\[1 em]x_{10} &= 0.3852+1.0393i\\[1 em]x_{11} &= 0.3852-1.0393i\\[1 em]x_{12} &= -0.8566+0.6324i\\[1 em]x_{13} &= -0.8566-0.6324i \end{aligned} $$

Step 1:

Write polynomial in descending order

$$ \begin{aligned} 5x^7-1+2x-8x^{13}+20x^3 & = 0\\[1 em] -8x^{13}+5x^7+20x^3+2x-1 & = 0 \end{aligned} $$

Step 2:

Polynomial $ -8x^{13}+5x^7+20x^3+2x-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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