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

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

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

Step 1:

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

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

Step 2:

Write polynomial in descending order

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

Step 3:

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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