Question

determine the equation of the slant asymptote of the rational function: $y = \frac{x^{2}-16}{x - 8}$
$y=-x + 8$
$y=x - 8$
$y=-x - 8$
$y=x + 8$

Explanation:

Step1: Perform polynomial long - division

Divide $x^{2}-16$ by $x - 8$. We know that $x^{2}-16=(x - 8)(x+8)+48$. So, $\frac{x^{2}-16}{x - 8}=\frac{(x - 8)(x + 8)+48}{x - 8}=x + 8+\frac{48}{x - 8}$.

Step2: Determine the slant asymptote

As $x\to\pm\infty$, the term $\frac{48}{x - 8}\to0$. The slant asymptote is given by the non - remainder part of the polynomial long - division result.

Answer:

$y=x + 8$