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

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 2 \end{aligned} $$

Step 1:

Write polynomial in descending order

$$ \begin{aligned} 2x^3-x^4 & = 0\\[1 em] -x^4+2x^3 & = 0 \end{aligned} $$

Step 2:

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

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

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-2 = 0 $

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

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