Find roots of polynomial $$ p(x) = -3x^7-x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 0.7211+0.4163i\\[1 em]x_3 &= 0.7211-0.4163i\\[1 em]x_4 &= 0.8327i\\[1 em]x_5 &= -0.8327i\\[1 em]x_6 &= -0.7211+0.4163i\\[1 em]x_7 &= -0.7211-0.4163i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ -x }$ from $ -3x^7-x $ and solve two separate equations:

$$ \begin{aligned} -3x^7-x & = 0\\[1 em] \color{blue}{ -x }\cdot ( 3x^6+1 ) & = 0 \\[1 em] \color{blue}{ -x = 0} ~~ \text{or} ~~ 3x^6+1 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ 3x^6+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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