Find roots of polynomial $$ p(x) = -16507500+542500x^3+77500x^4-232500x^5+387500x^6+x^8 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -1.8236\\[1 em]x_2 &= 1.8859\\[1 em]x_3 &= 1.0781+1.6826i\\[1 em]x_4 &= 1.0781-1.6826i\\[1 em]x_5 &= -0.8092+1.5643i\\[1 em]x_6 &= -0.8092-1.5643i\\[1 em]x_7 &= -0.3+622.495i\\[1 em]x_8 &= -0.3-622.495i \end{aligned} $$Explanation
Step 1:
Write polynomial in descending order
$$ \begin{aligned} -16507500+542500x^3+77500x^4-232500x^5+387500x^6+x^8 & = 0\\[1 em] x^8+387500x^6-232500x^5+77500x^4+542500x^3-16507500 & = 0 \end{aligned} $$Step 2:
Polynomial $ x^8+387500x^6-232500x^5+77500x^4+542500x^3-16507500 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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