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

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

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

Step 1:

Write polynomial in descending order

$$ \begin{aligned} x+x^2 & = 0\\[1 em] x^2+x & = 0 \end{aligned} $$

Step 2:

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

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

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

Step 3:

To find the second zero, solve equation $ x+1 = 0 $

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

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