Hello -

I have a 2-part question where I need to **solve the rational expression**: (x^5 + 4)/(x^3 - 1). First, I had a little trouble with the long division, getting the answer **x^2 with remainder (-x^2 + 4)**. But when I double-checked my answer by multiplying (x^3 - 1)[x^2 + (-x^2 + 4)/(x^3 - 1)], I couldn't reproduce the original rational expression. And second, does this result in a "non-linear" oblique asymptote, as in this case, x^2, indicating like some kind of "**parabolic asymptote**"? OR--are there only "linear" oblique asymptotes that occur when the power of the numerator is only 1 more than the power of the denominator?

Thanks

You can't "

**solve**" an expression. What was the actual wording of the problem? Are you to graph it?

Your

**remainder **is wrong. Please show your work. It's great that you checked it, but that check should also reveal why the remainder is wrong. Hint: sign errors are very common in this sort of work!

Technically, an asymptote has to be a line, and textbooks typically only consider linear asymptotes; but yes, y = x^2 is a

**curve** that your function asymptotically approaches. See

en.wikipedia.org

In analytic geometry, an asymptote (/ˈæsɪmptoʊt/) of a curve is a **line **such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity.

...

Let A : (a,b) → R2 be a parametric plane curve, in coordinates A(t) = (x(t),y(t)), and B be another (unparameterized) curve. Suppose, as before, that the curve A tends to infinity. The curve B is a **curvilinear **asymptote of A if the shortest distance from the point A(t) to a point on B tends to zero as t → b. Sometimes B is simply referred to as an asymptote of A, when there is no risk of confusion with **linear** asymptotes.[