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

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

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

Step 1:

Combine like terms:

$$ x^3 \color{blue}{-2x^2} \color{blue}{-x^2} = x^3 \color{blue}{-3x^2} $$

Step 2:

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

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

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

Step 3:

To find the last zero, solve equation $ x-3 = 0 $

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

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