Find roots of polynomial $$ p(x) = 5s^4+6s^3+2s^2+3s $$

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

$$ \begin{aligned}s_1 &= 0\\[1 em]s_2 &= -1.2604\\[1 em]s_3 &= 0.0302+0.6893i\\[1 em]s_4 &= 0.0302-0.6893i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ s }$ from $ 5s^4+6s^3+2s^2+3s $ and solve two separate equations:

$$ \begin{aligned} 5s^4+6s^3+2s^2+3s & = 0\\[1 em] \color{blue}{ s }\cdot ( 5s^3+6s^2+2s+3 ) & = 0 \\[1 em] \color{blue}{ s = 0} ~~ \text{or} ~~ 5s^3+6s^2+2s+3 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ 5s^3+6s^2+2s+3 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.

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