Find roots of polynomial $$ p(x) = 10+45x+\frac{4}{10}x-10x^2+45x^3+\frac{4}{10}x^3-9x^4-\frac{7}{10}x^4+4x^5 $$

The roots of polynomial $ p(x) $ are:

$$ \begin{aligned}x_1 &= -0.2025\\[1 em]x_2 &= 0.0961+1.0956i\\[1 em]x_3 &= 0.0961-1.0956i\\[1 em]x_4 &= 1.2177+2.9533i\\[1 em]x_5 &= 1.2177-2.9533i \end{aligned} $$

Step 1:

Get rid of fractions by multipling equation by $ \color{blue}{ 10 } $.

$$ \begin{aligned} 10+45x+\frac{4}{10}x-10x^2+45x^3+\frac{4}{10}x^3-9x^4-\frac{7}{10}x^4+4x^5 & = 0 ~~~ / \cdot \color{blue}{ 10 } \\[1 em] 100+450x+4x-100x^2+450x^3+4x^3-90x^4-7x^4+40x^5 & = 0 \end{aligned} $$

Step 2:

Combine like terms:

$$ 100+ \color{blue}{450x} + \color{blue}{4x} -100x^2+ \color{red}{450x^3} + \color{red}{4x^3} \color{green}{-90x^4} \color{green}{-7x^4} +40x^5 = \\ = 40x^5 \color{green}{-97x^4} + \color{red}{454x^3} -100x^2+ \color{blue}{454x} +100 $$

Step 3:

Polynomial $ 40x^5-97x^4+454x^3-100x^2+454x+100 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.

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