I apologise if this question is a little too vague (e.g., "meaningful"). I have included the soft-question tag for good measure.
Motivation:
This is motivated by one of those annoying memes on social media; effectively, it was
Solve $$\frac{x^5}{x+x}=8.$$
Of course, $x=\pm 2, \pm 2i$.
But some other people's solutions went like so:
Rewrite as $x^5-16x=0$ and factor: $x(x^2-4)(x^2+4)=0$, which gives the four solutions above plus a nonviable $x=0$.
The problem is not new to most here. The solution to divide the polynomials on the LHS first is superior.
But it got me thinking . . .
The Question:
Consider situations such as $$\frac{x^3}{x}=0\tag{1}$$ or more generally $$\frac{P(x)}{Q(x)}=R(x)\tag{$*$}$$ for (let's say real) polynomials $P,Q,R$, where $Q(x)$ can be zero for some $x$. Does allowing $0/0:=\bot$ as in wheel theory lead to meaningful solutions?
Thoughts:
It seems almost the point of wheel theory to give meaning to division by zero, right? But I haven't seen any of the theory around division of polynomials using $\bot$.
My current degree focuses on something tangential to algebraic varieties. I learnt that one can use similar techniques to define the dual number $\epsilon$ such that $\epsilon^2=0$ but $\epsilon\neq 0$. In that context, it seems sensible to say that $\pm \epsilon$ are solutions to $(1)$ (although dual numbers do not form a field, meaning $\epsilon^{-1}$ is undefined), but why not some other solution involving $\bot$ to "account for" the division by $x$ (in a nonstandard way)?
