Find roots of polynomial $$ p(x) = 24x^5+18x^4-x^3-23x-\frac{1}{21} $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -0.0021\\[1 em]x_2 &= -1.2606\\[1 em]x_3 &= 0.852\\[1 em]x_4 &= -0.1697+0.9292i\\[1 em]x_5 &= -0.1697-0.9292i \end{aligned} $$Explanation
Step 1:
Get rid of fractions by multipling equation by $ \color{blue}{ 21 } $.
$$ \begin{aligned} 24x^5+18x^4-x^3-23x-\frac{1}{21} & = 0 ~~~ / \cdot \color{blue}{ 21 } \\[1 em] 504x^5+378x^4-21x^3-483x-1 & = 0 \end{aligned} $$Step 2:
Polynomial $ 504x^5+378x^4-21x^3-483x-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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