Find roots of polynomial $$ p(x) = x^3+100x^2+4120x+4120x^{20} $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -0.9987\\[1 em]x_3 &= 0.6772+0.737i\\[1 em]x_4 &= 0.6772-0.737i\\[1 em]x_5 &= -0.5475+0.836i\\[1 em]x_6 &= -0.5475-0.836i\\[1 em]x_7 &= -0.2466+0.9688i\\[1 em]x_8 &= -0.2466-0.9688i\\[1 em]x_9 &= 0.8802+0.477i\\[1 em]x_{10} &= 0.8802-0.477i\\[1 em]x_{11} &= 0.4008+0.9167i\\[1 em]x_{12} &= 0.4008-0.9167i\\[1 em]x_{13} &= -0.7888+0.613i\\[1 em]x_{14} &= -0.7888-0.613i\\[1 em]x_{15} &= 0.0813+0.9968i\\[1 em]x_{16} &= 0.0813-0.9968i\\[1 em]x_{17} &= 0.9876+0.165i\\[1 em]x_{18} &= 0.9876-0.165i\\[1 em]x_{19} &= -0.9448+0.3239i\\[1 em]x_{20} &= -0.9448-0.3239i \end{aligned} $$

Step 1:

Write polynomial in descending order

$$ \begin{aligned} x^3+100x^2+4120x+4120x^{20} & = 0\\[1 em] 4120x^{20}+x^3+100x^2+4120x & = 0 \end{aligned} $$

Step 2:

Factor out $ \color{blue}{ x }$ from $ 4120x^{20}+x^3+100x^2+4120x $ and solve two separate equations:

$$ \begin{aligned} 4120x^{20}+x^3+100x^2+4120x & = 0\\[1 em] \color{blue}{ x }\cdot ( 4120x^{19}+x^2+100x+4120 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ 4120x^{19}+x^2+100x+4120 & = 0 \end{aligned} $$

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

Step 3:

Polynomial $ 4120x^{19}+x^2+100x+4120 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.

Polynomial Roots Calculator