Find roots of polynomial $$ p(x) = x+\frac{1}{2}x^2+\frac{1}{3}x^3+\frac{1}{4}x^4+\frac{1}{5}x^5+\frac{1}{6}x^6 $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -1.4623\\[1 em]x_3 &= 0.8051+1.1417i\\[1 em]x_4 &= 0.8051-1.1417i\\[1 em]x_5 &= -0.674+1.2838i\\[1 em]x_6 &= -0.674-1.2838i \end{aligned} $$

Step 1:

Get rid of fractions by multipling equation by $ \color{blue}{ 60 } $.

$$ \begin{aligned} x+\frac{1}{2}x^2+\frac{1}{3}x^3+\frac{1}{4}x^4+\frac{1}{5}x^5+\frac{1}{6}x^6 & = 0 ~~~ / \cdot \color{blue}{ 60 } \\[1 em] 60x+30x^2+20x^3+15x^4+12x^5+10x^6 & = 0 \end{aligned} $$

Step 2:

Write polynomial in descending order

$$ \begin{aligned} 60x+30x^2+20x^3+15x^4+12x^5+10x^6 & = 0\\[1 em] 10x^6+12x^5+15x^4+20x^3+30x^2+60x & = 0 \end{aligned} $$

Step 3:

Factor out $ \color{blue}{ x }$ from $ 10x^6+12x^5+15x^4+20x^3+30x^2+60x $ and solve two separate equations:

$$ \begin{aligned} 10x^6+12x^5+15x^4+20x^3+30x^2+60x & = 0\\[1 em] \color{blue}{ x }\cdot ( 10x^5+12x^4+15x^3+20x^2+30x+60 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ 10x^5+12x^4+15x^3+20x^2+30x+60 & = 0 \end{aligned} $$

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

Step 4:

Polynomial $ 10x^5+12x^4+15x^3+20x^2+30x+60 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.

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