Find roots of polynomial $$ p(x) = x^7-18x^6+123x^5-396x^4+612x^3-432x^2+112x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 0.7639\\[1 em]x_3 &= 0.7639\\[1 em]x_4 &= 1.5858\\[1 em]x_5 &= 4.4142\\[1 em]x_6 &= 5.2361\\[1 em]x_7 &= 5.2361 \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ x }$ from $ x^7-18x^6+123x^5-396x^4+612x^3-432x^2+112x $ and solve two separate equations:

$$ \begin{aligned} x^7-18x^6+123x^5-396x^4+612x^3-432x^2+112x & = 0\\[1 em] \color{blue}{ x }\cdot ( x^6-18x^5+123x^4-396x^3+612x^2-432x+112 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ x^6-18x^5+123x^4-396x^3+612x^2-432x+112 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ x^6-18x^5+123x^4-396x^3+612x^2-432x+112 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.

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