Find roots of polynomial $$ p(x) = 10001-26139x+26131x^2-3365x^3+1368x^4+5326x^5-2329x^6-3545x^7+2335x^8 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0.525+0.349i\\[1 em]x_2 &= 0.525-0.349i\\[1 em]x_3 &= -1.395+0.6759i\\[1 em]x_4 &= -1.395-0.6759i\\[1 em]x_5 &= 0.0515+1.2292i\\[1 em]x_6 &= 0.0515-1.2292i\\[1 em]x_7 &= 1.5777+0.6885i\\[1 em]x_8 &= 1.5777-0.6885i \end{aligned} $$Explanation
Step 1:
Write polynomial in descending order
$$ \begin{aligned} 10001-26139x+26131x^2-3365x^3+1368x^4+5326x^5-2329x^6-3545x^7+2335x^8 & = 0\\[1 em] 2335x^8-3545x^7-2329x^6+5326x^5+1368x^4-3365x^3+26131x^2-26139x+10001 & = 0 \end{aligned} $$Step 2:
Polynomial $ 2335x^8-3545x^7-2329x^6+5326x^5+1368x^4-3365x^3+26131x^2-26139x+10001 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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