Find roots of polynomial $$ p(x) = 54x^5+15x^4+8x^3 $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -\frac{ 5 }{ 36 }+\frac{\sqrt{ 167 }}{ 36 }i\\[1 em]x_3 &= -\frac{ 5 }{ 36 }- \frac{\sqrt{ 167 }}{ 36 }i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ x^3 }$ from $ 54x^5+15x^4+8x^3 $ and solve two separate equations:

$$ \begin{aligned} 54x^5+15x^4+8x^3 & = 0\\[1 em] \color{blue}{ x^3 }\cdot ( 54x^2+15x+8 ) & = 0 \\[1 em] \color{blue}{ x^3 = 0} ~~ \text{or} ~~ 54x^2+15x+8 & = 0 \end{aligned} $$

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

Step 2:

The solutions of $ 54x^2+15x+8 = 0 $ are: $ x = -\dfrac{ 5 }{ 36 }+\dfrac{\sqrt{ 167 }}{ 36 }i ~ \text{and} ~ x = -\dfrac{ 5 }{ 36 }-\dfrac{\sqrt{ 167 }}{ 36 }i$.

You can use step-by-step quadratic equation solver to see a detailed explanation on how to solve this quadratic equation.

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