Find roots of polynomial $$ p(x) = -27x^5-18x^4-3x^3 $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -\frac{ 1 }{ 3 } \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ -3x^3 }$ from $ -27x^5-18x^4-3x^3 $ and solve two separate equations:

$$ \begin{aligned} -27x^5-18x^4-3x^3 & = 0\\[1 em] \color{blue}{ -3x^3 }\cdot ( 9x^2+6x+1 ) & = 0 \\[1 em] \color{blue}{ -3x^3 = 0} ~~ \text{or} ~~ 9x^2+6x+1 & = 0 \end{aligned} $$

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

Step 2:

The solution of $ ~ 9x^2+6x+1 = 0 ~$ is $~ x = -\dfrac{ 1 }{ 3 } ~$.

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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