Question

Determine if the graph of the rational function has a slant asymptote. If it does, find the equation of the slant asymptote.


(2x^3+18x^2+47x+24)/(x^2+6x+5)

Answer 1

As x gets very large in magnitude this looks like
2 x^3/x^2
or in other words 2x
line with slope of 2
y = 2x

Answer 2

Oh, when x = 0
y = 24/5
so
y = 2 x + 24/5

Answer 3

So a long division to get

(2x^3 + 18x^2 + 47x + 24)/ (x^2 + 6x + 5) = 2x + 6 + (x-6)/(x^2+6x+5)

so as x get larger, (x-6)/(x^2+6x+5)
---> 0
and we are left with
y = 2x + 6

So the slant asymptote is y = 2x + 6