Question

find the equation of the slant asymptote of the rational function given below.
$f(x) = \frac{x^2 + 15x + 72}{x + 8}$

Explanation:

Step1: Perform polynomial long division

We divide the numerator \(x^{2}+15x + 72\) by the denominator \(x + 8\).
Using long division:
\(x^{2}+15x + 72=(x + 8)(x + 7)+16\)
So, \(\frac{x^{2}+15x + 72}{x + 8}=x + 7+\frac{16}{x + 8}\)

Step2: Determine the slant asymptote

As \(x\) approaches \(\pm\infty\), the term \(\frac{16}{x + 8}\) approaches \(0\). So the slant asymptote is the line \(y=x + 7\)

Answer:

The equation of the slant asymptote is \(y = x+7\)