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)
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
Oh, when x = 0
y = 24/5
so
y = 2 x + 24/5
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
To determine if the graph of a rational function has a slant asymptote, we need to check if the degree of the numerator is exactly one greater than the degree of the denominator.
In this case, the degree of the numerator is 3 and the degree of the denominator is 2. Since the degree of the numerator is not exactly one greater than the degree of the denominator, the graph does not have a slant asymptote.
When the degrees of the numerator and denominator do not match, the graph may have a slant asymptote. To find the equation of the slant asymptote, we can perform polynomial long division.
Let's divide the numerator (2x^3 + 18x^2 + 47x + 24) by the denominator (x^2 + 6x + 5):
2x
------
x^2 + 6x + 5 | 2x^3 + 18x^2 + 47x + 24
- (2x^3 + 12x^2 + 10x)
----------------------
6x^2 + 37x + 24
- (6x^2 + 36x + 30)
-----------------------
x - 6
The result is x - 6, which represents the slant asymptote. Therefore, the equation of the slant asymptote is y = x - 6.