This question already has an answer here:
I had this problem at school (we're just doing polynomial equations):
$$2 x^3 - x^2 - 3x = 0$$
I can see that $x = 0$ is a solution to this. But I divided left and right side by $x$, factorised, and didn't get the $0$ solution.
I can see I was wrong. But why is there this exception to the rule that "you can always do the same thing to the left and right side of an equation"?