Find roots of polynomial $$ p(x) = 4x^2-14+x^8 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -1.2843\\[1 em]x_2 &= 1.2843\\[1 em]x_3 &= 1.058+0.9232i\\[1 em]x_4 &= 1.058-0.9232i\\[1 em]x_5 &= -1.058+0.9232i\\[1 em]x_6 &= -1.058-0.9232i\\[1 em]x_7 &= 1.4777i\\[1 em]x_8 &= -1.4777i \end{aligned} $$Explanation
Step 1:
Write polynomial in descending order
$$ \begin{aligned} 4x^2-14+x^8 & = 0\\[1 em] x^8+4x^2-14 & = 0 \end{aligned} $$Step 2:
Polynomial $ x^8+4x^2-14 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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