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

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

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

Step 1:

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

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

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

Step 2:

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

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

Polynomial Roots Calculator