Find roots of polynomial $$ p(x) = 412x^{11}+4103x^{10}+18370x^9+48675x^8+84480x^7+100254x^6+82236x^5+45870x^4+16500x^3+3355x^2+242x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -0.1353\\[1 em]x_3 &= -1.6883\\[1 em]x_4 &= -0.4266+0.4923i\\[1 em]x_5 &= -0.4266-0.4923i\\[1 em]x_6 &= -0.8246+0.6977i\\[1 em]x_7 &= -0.8246-0.6977i\\[1 em]x_8 &= -1.2476+0.6555i\\[1 em]x_9 &= -1.2476-0.6555i\\[1 em]x_{10} &= -1.5688+0.3928i\\[1 em]x_{11} &= -1.5688-0.3928i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ x }$ from $ 412x^{11}+4103x^{10}+18370x^9+48675x^8+84480x^7+100254x^6+82236x^5+45870x^4+16500x^3+3355x^2+242x $ and solve two separate equations:

$$ \begin{aligned} 412x^{11}+4103x^{10}+18370x^9+48675x^8+84480x^7+100254x^6+82236x^5+45870x^4+16500x^3+3355x^2+242x & = 0\\[1 em] \color{blue}{ x }\cdot ( 412x^{10}+4103x^9+18370x^8+48675x^7+84480x^6+100254x^5+82236x^4+45870x^3+16500x^2+3355x+242 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ 412x^{10}+4103x^9+18370x^8+48675x^7+84480x^6+100254x^5+82236x^4+45870x^3+16500x^2+3355x+242 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ 412x^{10}+4103x^9+18370x^8+48675x^7+84480x^6+100254x^5+82236x^4+45870x^3+16500x^2+3355x+242 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.

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