Find roots of polynomial $$ p(x) = \frac{1}{2}x^2-\frac{1}{2}x^3 $$

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

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

Step 1:

Get rid of fractions by multipling equation by $ \color{blue}{ 2 } $.

$$ \begin{aligned} \frac{1}{2}x^2-\frac{1}{2}x^3 & = 0 ~~~ / \cdot \color{blue}{ 2 } \\[1 em] x^2-x^3 & = 0 \end{aligned} $$

Step 2:

Write polynomial in descending order

$$ \begin{aligned} x^2-x^3 & = 0\\[1 em] -x^3+x^2 & = 0 \end{aligned} $$

Step 3:

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

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

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

Step 4:

To find the last zero, solve equation $ x-1 = 0 $

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

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