Find roots of polynomial $$ p(x) = x^2+\frac{18}{5}x^{10}+\frac{72}{25}x^{10} $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 0.303+0.7314i\\[1 em]x_3 &= 0.303-0.7314i\\[1 em]x_4 &= -0.303+0.7314i\\[1 em]x_5 &= -0.303-0.7314i\\[1 em]x_6 &= -0.7314+0.303i\\[1 em]x_7 &= -0.7314-0.303i\\[1 em]x_8 &= 0.7314+0.303i\\[1 em]x_9 &= 0.7314-0.303i \end{aligned} $$

Step 1:

Get rid of fractions by multipling equation by $ \color{blue}{ 25 } $.

$$ \begin{aligned} x^2+\frac{18}{5}x^{10}+\frac{72}{25}x^{10} & = 0 ~~~ / \cdot \color{blue}{ 25 } \\[1 em] 25x^2+90x^{10}+72x^{10} & = 0 \end{aligned} $$

Step 2:

Combine like terms:

$$ 25x^2+ \color{blue}{90x^{10}} + \color{blue}{72x^{10}} = \color{blue}{162x^{10}} +25x^2 $$

Step 3:

Factor out $ \color{blue}{ x^2 }$ from $ 162x^{10}+25x^2 $ and solve two separate equations:

$$ \begin{aligned} 162x^{10}+25x^2 & = 0\\[1 em] \color{blue}{ x^2 }\cdot ( 162x^8+25 ) & = 0 \\[1 em] \color{blue}{ x^2 = 0} ~~ \text{or} ~~ 162x^8+25 & = 0 \end{aligned} $$

One solution is $ \color{blue}{ x = 0 } $. Use second equation to find the remaining roots.

Step 4:

Polynomial $ 162x^8+25 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.

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