Find roots of polynomial $$ p(x) = 8x^4+5x^5 $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -\frac{ 8 }{ 5 } \end{aligned} $$

Step 1:

Write polynomial in descending order

$$ \begin{aligned} 8x^4+5x^5 & = 0\\[1 em] 5x^5+8x^4 & = 0 \end{aligned} $$

Step 2:

Factor out $ \color{blue}{ x^4 }$ from $ 5x^5+8x^4 $ and solve two separate equations:

$$ \begin{aligned} 5x^5+8x^4 & = 0\\[1 em] \color{blue}{ x^4 }\cdot ( 5x+8 ) & = 0 \\[1 em] \color{blue}{ x^4 = 0} ~~ \text{or} ~~ 5x+8 & = 0 \end{aligned} $$

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+8 = 0 $

$$ \begin{aligned} 5x+8 & = 0 \\[1 em] 5 \cdot x & = -8 \\[1 em] x & = - \frac{ 8 }{ 5 } \end{aligned} $$

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