Find roots of polynomial $$ p(x) = x^8-4x^3+2x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 0.7256\\[1 em]x_3 &= -0.6934\\[1 em]x_4 &= 1.2147\\[1 em]x_5 &= -1.0174+0.834i\\[1 em]x_6 &= -1.0174-0.834i\\[1 em]x_7 &= 0.3939+1.3176i\\[1 em]x_8 &= 0.3939-1.3176i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ x }$ from $ x^8-4x^3+2x $ and solve two separate equations:

$$ \begin{aligned} x^8-4x^3+2x & = 0\\[1 em] \color{blue}{ x }\cdot ( x^7-4x^2+2 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ x^7-4x^2+2 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ x^7-4x^2+2 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.

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