Find roots of polynomial $$ p(x) = 5x^3-29x^2+20x^2 $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= \frac{ 9 }{ 5 } \end{aligned} $$

Step 1:

Combine like terms:

$$ 5x^3 \color{blue}{-29x^2} + \color{blue}{20x^2} = 5x^3 \color{blue}{-9x^2} $$

Step 2:

Factor out $ \color{blue}{ x^2 }$ from $ 5x^3-9x^2 $ and solve two separate equations:

$$ \begin{aligned} 5x^3-9x^2 & = 0\\[1 em] \color{blue}{ x^2 }\cdot ( 5x-9 ) & = 0 \\[1 em] \color{blue}{ x^2 = 0} ~~ \text{or} ~~ 5x-9 & = 0 \end{aligned} $$

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

Step 3:

To find the last zero, solve equation $ 5x-9 = 0 $

$$ \begin{aligned} 5x-9 & = 0 \\[1 em] 5 \cdot x & = 9 \\[1 em] x & = \frac{ 9 }{ 5 } \end{aligned} $$

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