Find roots of polynomial $$ p(x) = 5x^7-2x^8-x^5+4 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -0.9486\\[1 em]x_2 &= 2.4187\\[1 em]x_3 &= 0.9531+0.4751i\\[1 em]x_4 &= 0.9531-0.4751i\\[1 em]x_5 &= -0.6157+0.6916i\\[1 em]x_6 &= -0.6157-0.6916i\\[1 em]x_7 &= 0.1776+0.93i\\[1 em]x_8 &= 0.1776-0.93i \end{aligned} $$Explanation
Step 1:
Write polynomial in descending order
$$ \begin{aligned} 5x^7-2x^8-x^5+4 & = 0\\[1 em] -2x^8+5x^7-x^5+4 & = 0 \end{aligned} $$Step 2:
Polynomial $ -2x^8+5x^7-x^5+4 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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