Find roots of polynomial $$ p(x) = 2232+81590x+965770x^2-3259660x^3-601906x^4+8387290x^5-3667880x^6-5582480x^7+3676840x^8 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -0.0365+0.023i\\[1 em]x_2 &= -0.0365-0.023i\\[1 em]x_3 &= -0.8596+0.013i\\[1 em]x_4 &= -0.8596-0.013i\\[1 em]x_5 &= 0.6828+0.0254i\\[1 em]x_6 &= 0.6828-0.0254i\\[1 em]x_7 &= 0.9724+0.0231i\\[1 em]x_8 &= 0.9724-0.0231i \end{aligned} $$Explanation
Step 1:
Write polynomial in descending order
$$ \begin{aligned} 2232+81590x+965770x^2-3259660x^3-601906x^4+8387290x^5-3667880x^6-5582480x^7+3676840x^8 & = 0\\[1 em] 3676840x^8-5582480x^7-3667880x^6+8387290x^5-601906x^4-3259660x^3+965770x^2+81590x+2232 & = 0 \end{aligned} $$Step 2:
Polynomial $ 3676840x^8-5582480x^7-3667880x^6+8387290x^5-601906x^4-3259660x^3+965770x^2+81590x+2232 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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