Find roots of polynomial $$ p(x) = x^6+\frac{1289}{10}x^5+2555x^4+\frac{93781}{5}x^3+\frac{264708}{5}x^2+\frac{241704}{5}x+40284 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -4.6145\\[1 em]x_2 &= -106.5244\\[1 em]x_3 &= -0.4107+0.9653i\\[1 em]x_4 &= -0.4107-0.9653i\\[1 em]x_5 &= -8.4699+1.6541i\\[1 em]x_6 &= -8.4699-1.6541i \end{aligned} $$Explanation
Step 1:
Get rid of fractions by multipling equation by $ \color{blue}{ 10 } $.
$$ \begin{aligned} x^6+\frac{1289}{10}x^5+2555x^4+\frac{93781}{5}x^3+\frac{264708}{5}x^2+\frac{241704}{5}x+40284 & = 0 ~~~ / \cdot \color{blue}{ 10 } \\[1 em] 10x^6+1289x^5+25550x^4+187562x^3+529416x^2+483408x+402840 & = 0 \end{aligned} $$Step 2:
Polynomial $ 10x^6+1289x^5+25550x^4+187562x^3+529416x^2+483408x+402840 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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