Rewrite the function using either Polynomial Long Division or the method below.
f(x) = (x^2-5x+6)/(x+4)f(x)=x2−5x+6x+4
By long division of polynomials (or, in this case synthetic division will also work), we can get:
f(x) = x-9 + 42/(x+4)f(x)=x−9+42x+4.
So y = x-9y=x−9 is an asymptote.
Long division is a but tedious (difficult) to format nicely here, so I'll show another way to think of this.
x(x+4) = x^2 +4xx(x+4)=x2+4x Use this to rewrite:
f(x) = (x^2+4x-9x+6)/(x+4)f(x)=x2+4x−9x+6x+4
Notice that we replaced the -5x−5x by 4x-9x4x−9x. This is done so that we can group and reduce.
We have:
f(x) = (x^2+4x-9x+6)/(x+4) = (x^2+4x)/(x+4) + (-9x+6)/(x+4)f(x)=x2+4x−9x+6x+4=x2+4xx+4+−9x+6x+4
= x + (-9x+6)/(x+4)=x+−9x+6x+4
Now, we'll use -9(x+4) = -9x-36−9(x+4)=−9x−36 to get:
f(x) = x + (-9x-36+42)/(x+4)f(x)=x+−9x−36+42x+4 (Replacing +6+6 with -36+42−36+42 to get:
f(x) = x + (-9x-36)/(x+4) +42/(x+4)f(x)=x+−9x−36x+4+42x+4
= x-9 + 42/(x+4)=x−9+42x+4
Writing:
f(x) = x-9 + 42/(x+4)f(x)=x−9+42x+4 we can see that the difference is:
f(x) - (x-9) = 42/(x+4)f(x)−(x−9)=42x+4
As xrarroox→∞ and as xrarr -oox→−∞, this difference goes to 00, so
the line y = x-9y=x−9 is an asymptote for f(x)f(x) (on both sides).
