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

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -\frac{ 1 }{ 2 }+\frac{\sqrt{ 3 }}{ 2 }i\\[1 em]x_3 &= -\frac{ 1 }{ 2 }- \frac{\sqrt{ 3 }}{ 2 }i \end{aligned} $$

Step 1:

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

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

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

Step 2:

The solutions of $ x^2+x+1 = 0 $ are: $ x = -\dfrac{ 1 }{ 2 }+\dfrac{\sqrt{ 3 }}{ 2 }i ~ \text{and} ~ x = -\dfrac{ 1 }{ 2 }-\dfrac{\sqrt{ 3 }}{ 2 }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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