Find roots of polynomial $$ p(x) = x^5+5x^4+10x^3+10x^2+5x+\frac{7}{2} $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -2.2011\\[1 em]x_2 &= -0.0283+0.706i\\[1 em]x_3 &= -0.0283-0.706i\\[1 em]x_4 &= -1.3712+1.1423i\\[1 em]x_5 &= -1.3712-1.1423i \end{aligned} $$Explanation
Step 1:
Get rid of fractions by multipling equation by $ \color{blue}{ 2 } $.
$$ \begin{aligned} x^5+5x^4+10x^3+10x^2+5x+\frac{7}{2} & = 0 ~~~ / \cdot \color{blue}{ 2 } \\[1 em] 2x^5+10x^4+20x^3+20x^2+10x+7 & = 0 \end{aligned} $$Step 2:
Polynomial $ 2x^5+10x^4+20x^3+20x^2+10x+7 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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