Given a polynomial $P$, write $[P]$ for the homogeneous part of highest degree. For example:
$$[3x^2+4xy-y^2+17x-2y+5] = 3x^2+4xy-y^2$$
Question 0. Is there accepted notation for this?
Now define a function as follows. $$\mathbb{R}(x)_{\neq 0} \times \mathbb{R}(x)_{\neq 0} \xrightarrow{} \mathbb{R}(x)_{\neq 0}$$ $$P,Q \mapsto \frac{[P]}{[Q]}$$
Question 1. Is there an accepted name for this "pseudodivision" operation? I don't necessarily think it needs it's own notation, but having a word or phrase for this would be useful for communication.
By the way, there's at least two ways pseudodivision shows up. Firstly, it shows up naturally when analyzing the behavior of functions near infinity. For example, the pseudoquotient $$\frac{[3x^3+x-1]}{[-5x+1]} = -\frac{3}{5} x^2$$ indicates that the function $\frac{3x^3+x-1}{-5x+1}$ approaches $-\infty$ as $x$ goes to $+\infty$. Notice that, unlike with big-$O$ notation, we've held onto information about how fast the quotient is growing (which is sometimes desirable and sometimes not). In general, we have $$\mathrm{lim}_{x \rightarrow \infty}\frac{P}{Q} = \mathrm{lim}_{x \rightarrow \infty} \frac{[P]}{[Q]}$$ as long as $P$ and $Q$ are non-zero.
Personally, the reason I'm interested in pseudodivision thing is to teach polynomial long division better. To compute $P/Q$, create three columns. Write down $P$ in the first column, compute $\frac{[P]}{[Q]}$ and put this in the second column, then multiply by $Q$ and put this in the third column. Finally, start a new row by computing the first column minus the last column. If you repeat this until the degree of $P$ is strictly less than the degree of $Q$, the sum of the terms in the middle column gives you the quotient, and the bottom term in the first column is the remainder. Long story short, I think that first getting students to compute $[P]/[Q]$ for a couple of different polynomials and having a word like "pseudodivision" to refer to this operation could make teaching this material a bit easier, because that way there'll be less gesturing vigorously and more saying what I actually mean on my part.