Find roots of polynomial $$ p(x) = x+\frac{1}{2}x^2+\frac{1}{3}x^3+\frac{1}{4}x^4+\frac{1}{5}x^5+\frac{1}{6}x^6+\frac{1}{7}x^7 $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 0.9265+0.9755i\\[1 em]x_3 &= 0.9265-0.9755i\\[1 em]x_4 &= -0.2481+1.3721i\\[1 em]x_5 &= -0.2481-1.3721i\\[1 em]x_6 &= -1.2618+0.6303i\\[1 em]x_7 &= -1.2618-0.6303i \end{aligned} $$

Step 1:

Get rid of fractions by multipling equation by $ \color{blue}{ 420 } $.

$$ \begin{aligned} x+\frac{1}{2}x^2+\frac{1}{3}x^3+\frac{1}{4}x^4+\frac{1}{5}x^5+\frac{1}{6}x^6+\frac{1}{7}x^7 & = 0 ~~~ / \cdot \color{blue}{ 420 } \\[1 em] 420x+210x^2+140x^3+105x^4+84x^5+70x^6+60x^7 & = 0 \end{aligned} $$

Step 2:

Write polynomial in descending order

$$ \begin{aligned} 420x+210x^2+140x^3+105x^4+84x^5+70x^6+60x^7 & = 0\\[1 em] 60x^7+70x^6+84x^5+105x^4+140x^3+210x^2+420x & = 0 \end{aligned} $$

Step 3:

Factor out $ \color{blue}{ x }$ from $ 60x^7+70x^6+84x^5+105x^4+140x^3+210x^2+420x $ and solve two separate equations:

$$ \begin{aligned} 60x^7+70x^6+84x^5+105x^4+140x^3+210x^2+420x & = 0\\[1 em] \color{blue}{ x }\cdot ( 60x^6+70x^5+84x^4+105x^3+140x^2+210x+420 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ 60x^6+70x^5+84x^4+105x^3+140x^2+210x+420 & = 0 \end{aligned} $$

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

Step 4:

Polynomial $ 60x^6+70x^5+84x^4+105x^3+140x^2+210x+420 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.

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