Find roots of polynomial $$ p(x) = 6x^7-2x^6+4x^5+2x^4+7x^3+6x^2+18x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 1.0482+0.8521i\\[1 em]x_3 &= 1.0482-0.8521i\\[1 em]x_4 &= -0.8909+0.6305i\\[1 em]x_5 &= -0.8909-0.6305i\\[1 em]x_6 &= 0.0094+1.1748i\\[1 em]x_7 &= 0.0094-1.1748i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ x }$ from $ 6x^7-2x^6+4x^5+2x^4+7x^3+6x^2+18x $ and solve two separate equations:

$$ \begin{aligned} 6x^7-2x^6+4x^5+2x^4+7x^3+6x^2+18x & = 0\\[1 em] \color{blue}{ x }\cdot ( 6x^6-2x^5+4x^4+2x^3+7x^2+6x+18 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ 6x^6-2x^5+4x^4+2x^3+7x^2+6x+18 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ 6x^6-2x^5+4x^4+2x^3+7x^2+6x+18 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.

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