Find roots of polynomial $$ p(x) = 2x^6+x^5 $$

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

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

Step 1:

Factor out $ \color{blue}{ x^5 }$ from $ 2x^6+x^5 $ and solve two separate equations:

$$ \begin{aligned} 2x^6+x^5 & = 0\\[1 em] \color{blue}{ x^5 }\cdot ( 2x+1 ) & = 0 \\[1 em] \color{blue}{ x^5 = 0} ~~ \text{or} ~~ 2x+1 & = 0 \end{aligned} $$

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

Step 2:

To find the last zero, solve equation $ 2x+1 = 0 $

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

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