Find roots of polynomial $$ p(x) = p^4-p^5-\frac{1}{5} $$
Answer
The roots of polynomial $ p(p) $ are:
$$ \begin{aligned}p_1 &= -0.5951\\[1 em]p_2 &= -0.0827+0.6269i\\[1 em]p_3 &= -0.0827-0.6269i\\[1 em]p_4 &= 0.8803+0.2565i\\[1 em]p_5 &= 0.8803-0.2565i \end{aligned} $$Explanation
Step 1:
Get rid of fractions by multipling equation by $ \color{blue}{ 5 } $.
$$ \begin{aligned} p^4-p^5-\frac{1}{5} & = 0 ~~~ / \cdot \color{blue}{ 5 } \\[1 em] 5p^4-5p^5-1 & = 0 \end{aligned} $$Step 2:
Write polynomial in descending order
$$ \begin{aligned} 5p^4-5p^5-1 & = 0\\[1 em] -5p^5+5p^4-1 & = 0 \end{aligned} $$Step 3:
Polynomial $ -5p^5+5p^4-1 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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