Question

determine the equation of the slant asymptote of the rational function: $y = \frac{x^{2}-12x + 8}{x - 9}$
$y=x - 21$
$y=-x - 21$
$y=x - 3$
$y=-x - 3$

Explanation:

Step1: Perform polynomial long - division

Divide $x^{2}-12x + 8$ by $x - 9$.
\[

$$\begin{align*} x^{2}-12x + 8&=(x - 9)(x-3)-19 \end{align*}$$

\]
So, $\frac{x^{2}-12x + 8}{x - 9}=x - 3-\frac{19}{x - 9}$.

Step2: Determine the slant asymptote

As $x\to\pm\infty$, the term $\frac{-19}{x - 9}\to0$. The slant asymptote is given by the quotient of the polynomial long - division.

Answer:

$y=x - 3$