Question

determine the equation of the slant asymptote of the rational function: $y = \frac{x^{2}+7x - 5}{x + 2}$
$y=x + 5$
$y=-x + 5$
$y=x + 9$
$y=-x + 9$

Explanation:

Step1: Perform polynomial long - division

Divide $x^{2}+7x - 5$ by $x + 2$.
\[

$$\begin{align*} x^{2}+7x-5&=(x + 2)(x+5)-15 \end{align*}$$

\]
So, $\frac{x^{2}+7x - 5}{x + 2}=x + 5-\frac{15}{x + 2}$.

Step2: Determine the slant asymptote

As $x\to\pm\infty$, the term $\frac{-15}{x + 2}\to0$. The equation of the slant asymptote is given by the non - remainder part of the quotient.

Answer:

A. $y=x + 5$