Question

determine the equation of the slant asymptote of the rational function: $y = \frac{x^{2}-5x + 4}{x + 3}$
$y=x - 2$
$y=x + 8$
$y=x - 8$
$y=x + 2$

Explanation:

Step1: Perform polynomial long - division

Divide $x^{2}-5x + 4$ by $x + 3$. We know that $x^{2}-5x + 4=(x + 3)(x-8)+28$. The quotient of $\frac{x^{2}-5x + 4}{x + 3}$ using long - division of polynomials: $x-8+\frac{28}{x + 3}$.

Step2: Determine the slant asymptote

As $x\to\pm\infty$, the term $\frac{28}{x + 3}\to0$. The equation of the slant asymptote is given by the quotient of the polynomial long - division (excluding the remainder term). So the equation of the slant asymptote is $y=x - 8$.

Answer:

C. $y=x - 8$