Find roots of polynomial $$ p(x) = -4x^3+2x^2+x^6-8x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 1.7595\\[1 em]x_3 &= 0.3147+1.1574i\\[1 em]x_4 &= 0.3147-1.1574i\\[1 em]x_5 &= -1.1944+1.3168i\\[1 em]x_6 &= -1.1944-1.3168i \end{aligned} $$

Step 1:

Write polynomial in descending order

$$ \begin{aligned} -4x^3+2x^2+x^6-8x & = 0\\[1 em] x^6-4x^3+2x^2-8x & = 0 \end{aligned} $$

Step 2:

Factor out $ \color{blue}{ x }$ from $ x^6-4x^3+2x^2-8x $ and solve two separate equations:

$$ \begin{aligned} x^6-4x^3+2x^2-8x & = 0\\[1 em] \color{blue}{ x }\cdot ( x^5-4x^2+2x-8 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ x^5-4x^2+2x-8 & = 0 \end{aligned} $$

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

Step 3:

Polynomial $ x^5-4x^2+2x-8 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.

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