Division of Polynomials and Slant Asymptotes

Division of Polynomials and Slant Asymptotes

Here is more detail about long run behavior of rational functions when the degree of the numerator is greater than the degree of the denominator.

As the book points out (see p. 193 in the textbook), the long run behavior of rational functions mimics the ratio of the leading terms of the numerator and denominator function.

Example:  [pic 1] -- the long run behavior of this rational function is similar to [pic 2].  That is, it would look like a cubic.

But which cubic?  

To find the exact answer, you need to do a little long division.  

Reminder:  Write the improper fraction [pic 3]as a mixed number.

Do you remember what to do?  In the United States, you probably use the “gazinta” method – divide 17 by 6:[pic 4]

[pic 5]                        

We’ll use the same technique to find the specific long run behavior of rational functions.  If you learned a different algorithm for long division, the same ideas should apply.

[pic 6]

To find the specific function [pic 7] that [pic 8] is asymptotic to, we will perform long division.

Example:  Consider[pic 9].  Find [pic 10]so that [pic 11]is asymptotic to[pic 12].

Solution: Divide, just as we would with numbers.  It’s helpful to line up powers of x:

[pic 13]

[pic 14]        

Then we can write[pic 15]. What happens to this function in the long run?  The rational bit that we built out of the remainder has degree of the numerator less than the degree of the denominator, so it tends to zero.  As x gets far from zero in either direction, [pic 16] will look like[pic 17].  So [pic 18]is asymptotic to[pic 19].  Since this is a linear function, we can say that [pic 20]is an asymptote for g(x).  This asymptote is neither horizontal nor vertical; it is a slant asymptote or oblique asymptote.