Since dividing $x=x^6$ by $x$ gives $1=x^5$, how can I get to $x=0$ as a root? this might sound like a stupid question,
bear with me it probably is.

I know the solutions for $x=x^6$ are 1 and 0.
Now, since $1 \cdot x = 1 \cdot x^6 $ and it follows $ 1 \cdot x = 1 \cdot x \cdot x^5$ and I divide both sides by $ 1 \cdot x$ to get $ 1=x^5 $, how can I find $x=0$?

Thank you so much in advance.

You find $x=0$ by using the property that if $A\cdot B=0$, and $A,B$ are real numbers (or elements of a field), then $A=0$ or $B=0$.
Write $x=x^6\iff x^6-x=0\iff x(x^5-1)=0$
Either $x^5-1=0$ or $x=0$. The last thing shows that $x=0$ is a solution.
