Find roots of polynomial $$ p(x) = 54x^5+5x^4+36x^3+6x^2 $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -0.1638\\[1 em]x_3 &= 0.0356+0.8228i\\[1 em]x_4 &= 0.0356-0.8228i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ x^2 }$ from $ 54x^5+5x^4+36x^3+6x^2 $ and solve two separate equations:

$$ \begin{aligned} 54x^5+5x^4+36x^3+6x^2 & = 0\\[1 em] \color{blue}{ x^2 }\cdot ( 54x^3+5x^2+36x+6 ) & = 0 \\[1 em] \color{blue}{ x^2 = 0} ~~ \text{or} ~~ 54x^3+5x^2+36x+6 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ 54x^3+5x^2+36x+6 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.

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