Find roots of polynomial $$ p(x) = -35x^6+54x^5+6x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 1.571\\[1 em]x_3 &= 0.387+0.47i\\[1 em]x_4 &= 0.387-0.47i\\[1 em]x_5 &= -0.4011+0.3654i\\[1 em]x_6 &= -0.4011-0.3654i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ -x }$ from $ -35x^6+54x^5+6x $ and solve two separate equations:

$$ \begin{aligned} -35x^6+54x^5+6x & = 0\\[1 em] \color{blue}{ -x }\cdot ( 35x^5-54x^4-6 ) & = 0 \\[1 em] \color{blue}{ -x = 0} ~~ \text{or} ~~ 35x^5-54x^4-6 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ 35x^5-54x^4-6 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.

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