Find roots of polynomial $$ p(x) = x^6-7x^5 $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 7 \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ x^5 }$ from $ x^6-7x^5 $ and solve two separate equations:

$$ \begin{aligned} x^6-7x^5 & = 0\\[1 em] \color{blue}{ x^5 }\cdot ( x-7 ) & = 0 \\[1 em] \color{blue}{ x^5 = 0} ~~ \text{or} ~~ x-7 & = 0 \end{aligned} $$

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

Step 2:

To find the last zero, solve equation $ x-7 = 0 $

$$ \begin{aligned} x-7 & = 0 \\[1 em] x & = 7 \end{aligned} $$

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