Find roots of polynomial $$ p(x) = x^6-2x^3+5x^2+12x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -1.2304\\[1 em]x_3 &= 1.4379+1.0251i\\[1 em]x_4 &= 1.4379-1.0251i\\[1 em]x_5 &= -0.8227+1.5654i\\[1 em]x_6 &= -0.8227-1.5654i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ x }$ from $ x^6-2x^3+5x^2+12x $ and solve two separate equations:

$$ \begin{aligned} x^6-2x^3+5x^2+12x & = 0\\[1 em] \color{blue}{ x }\cdot ( x^5-2x^2+5x+12 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ x^5-2x^2+5x+12 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ x^5-2x^2+5x+12 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.

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