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.
 A: Whenever you divide by a variable quantity, you have to pause and think, "Either this is zero or not.  If it's not, I can divide by it.  If it is, that's another case."
So if $x$ is not $0$, you can divide by it to get $1=x^5$.  If it is zero, then, well, $x=0$.
But most folks would proceed like this:  $0 = x^6-x = x(x^5-1)$  and one of the last factors equals $0$.  So either $x=0$  or $x^5-1 = 0$.
A: Don't divide. Factorise:
$x=x^6$
$x-x^6=0$
$x(1-x^5)=0$
Either $x=0$ or $1-x^5=0 \Rightarrow x=1$
A: 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.
A: As said, you don't divide, but you factorize. Factorization gets all the solutions. Division, especially by $x$, as shown here:
$$ x = x^6 $$
then
$$ \frac{x}{x} = \frac{x^6}{x} $$
doesn't follow. In the first case, $x = 0$ is a solution, but then, in this case, $x = 0$ gets "washed away", because in this new equation, both sides evaluate to $\frac{0}{0}$, which is indeterminate.
A: You cannot divide by $0$, it is not allowed. If you have $x=x^6$, it is not necessarily equal to $1=x^5$ if $x$ can be $0$. Take this example of $0x=0y$. If you divide both sides by $0$, you get $x=y$ even though $x$ doesn't have to equal $y$ for $0x=0y$ to be true. Likewise, if you let $x=0$, $x=x^6$ can be rewritten as $0\times1 = 0\times x^5$, and if you divide both sides by $0$, you get $1=x^5$. As we have shown before, $1=x^5$ isn't necessarily true because $x$ can be $0$, according to the original equation.
