Find roots of polynomial $$ p(x) = x^5+\frac{5}{2}x^4-2x^3-6x^2+\frac{1}{2}x+2 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0.618\\[1 em]x_2 &= -0.655\\[1 em]x_3 &= -1.618\\[1 em]x_4 &= 1.3753\\[1 em]x_5 &= -2.2203 \end{aligned} $$Explanation
Step 1:
Get rid of fractions by multipling equation by $ \color{blue}{ 2 } $.
$$ \begin{aligned} x^5+\frac{5}{2}x^4-2x^3-6x^2+\frac{1}{2}x+2 & = 0 ~~~ / \cdot \color{blue}{ 2 } \\[1 em] 2x^5+5x^4-4x^3-12x^2+x+4 & = 0 \end{aligned} $$Step 2:
Polynomial $ 2x^5+5x^4-4x^3-12x^2+x+4 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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