Find roots of polynomial $$ p(x) = x^2-\frac{1}{2}x^5-1 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -0.8681\\[1 em]x_2 &= -0.5854+1.2767i\\[1 em]x_3 &= -0.5854-1.2767i\\[1 em]x_4 &= 1.0194+0.3589i\\[1 em]x_5 &= 1.0194-0.3589i \end{aligned} $$Explanation
Step 1:
Get rid of fractions by multipling equation by $ \color{blue}{ 2 } $.
$$ \begin{aligned} x^2-\frac{1}{2}x^5-1 & = 0 ~~~ / \cdot \color{blue}{ 2 } \\[1 em] 2x^2-x^5-2 & = 0 \end{aligned} $$Step 2:
Write polynomial in descending order
$$ \begin{aligned} 2x^2-x^5-2 & = 0\\[1 em] -x^5+2x^2-2 & = 0 \end{aligned} $$Step 3:
Polynomial $ -x^5+2x^2-2 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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