Polynomial Long Division & Oblique (non-linear?) Asymptote

[jacktosh]

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

 

[Dr.Peterson]

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

Asymptote - Wikipedia

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.[​