Find roots of polynomial $$ p(x) = 1+x+x^2+x^3+x^4 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0.309+0.9511i\\[1 em]x_2 &= 0.309-0.9511i\\[1 em]x_3 &= -0.809+0.5878i\\[1 em]x_4 &= -0.809-0.5878i \end{aligned} $$Explanation
Step 1:
Write polynomial in descending order
$$ \begin{aligned} 1+x+x^2+x^3+x^4 & = 0\\[1 em] x^4+x^3+x^2+x+1 & = 0 \end{aligned} $$Step 2:
Polynomial $ x^4+x^3+x^2+x+1 $ has no rational roots that can be found using Rational Root Test, so the roots were found using quartic formulas.
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