Find roots of polynomial $$ p(x) = 1000x+1000x^2+2000x^2+20x^3+10x^3+x^4 $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -0.3344\\[1 em]x_3 &= -14.8328+52.6314i\\[1 em]x_4 &= -14.8328-52.6314i \end{aligned} $$

Step 1:

Combine like terms:

$$ 1000x+ \color{blue}{1000x^2} + \color{blue}{2000x^2} + \color{red}{20x^3} + \color{red}{10x^3} +x^4 = x^4+ \color{red}{30x^3} + \color{blue}{3000x^2} +1000x $$

Step 2:

Factor out $ \color{blue}{ x }$ from $ x^4+30x^3+3000x^2+1000x $ and solve two separate equations:

$$ \begin{aligned} x^4+30x^3+3000x^2+1000x & = 0\\[1 em] \color{blue}{ x }\cdot ( x^3+30x^2+3000x+1000 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ x^3+30x^2+3000x+1000 & = 0 \end{aligned} $$

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

Step 3:

Polynomial $ x^3+30x^2+3000x+1000 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.

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