Find roots of polynomial $$ p(x) = 1+u^2-u^5 $$
Answer
The roots of polynomial $ p(u) $ are:
$$ \begin{aligned}u_1 &= 1.1939\\[1 em]u_2 &= 0.1546+0.8281i\\[1 em]u_3 &= 0.1546-0.8281i\\[1 em]u_4 &= -0.7515+0.7846i\\[1 em]u_5 &= -0.7515-0.7846i \end{aligned} $$Explanation
Step 1:
Write polynomial in descending order
$$ \begin{aligned} 1+u^2-u^5 & = 0\\[1 em] -u^5+u^2+1 & = 0 \end{aligned} $$Step 2:
Polynomial $ -u^5+u^2+1 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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