Find roots of polynomial $$ p(x) = z^6+11z^5 $$

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

$$ \begin{aligned}z_1 &= 0\\[1 em]z_2 &= -11 \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ z^5 }$ from $ z^6+11z^5 $ and solve two separate equations:

$$ \begin{aligned} z^6+11z^5 & = 0\\[1 em] \color{blue}{ z^5 }\cdot ( z+11 ) & = 0 \\[1 em] \color{blue}{ z^5 = 0} ~~ \text{or} ~~ z+11 & = 0 \end{aligned} $$

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

Step 2:

To find the last zero, solve equation $ z+11 = 0 $

$$ \begin{aligned} z+11 & = 0 \\[1 em] z & = -11 \end{aligned} $$

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