Find roots of polynomial $$ p(x) = 12x^7-14x^2-16x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -0.8406\\[1 em]x_3 &= 1.1808\\[1 em]x_4 &= 0.4753+1.0409i\\[1 em]x_5 &= 0.4753-1.0409i\\[1 em]x_6 &= -0.6454+0.7807i\\[1 em]x_7 &= -0.6454-0.7807i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ 2x }$ from $ 12x^7-14x^2-16x $ and solve two separate equations:

$$ \begin{aligned} 12x^7-14x^2-16x & = 0\\[1 em] \color{blue}{ 2x }\cdot ( 6x^6-7x-8 ) & = 0 \\[1 em] \color{blue}{ 2x = 0} ~~ \text{or} ~~ 6x^6-7x-8 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ 6x^6-7x-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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