Question

the rational function $r$ is given by $r(x)=\frac{x^{3}+8x^{2}+17x + 10}{x^{2}+2x}=\frac{(x + 1)(x + 2)(x + 5)}{x^{2}+2x}$. which of the following gives equations for all the horizontal asymptotes, vertical asymptotes, and slant asymptotes of the graph of $r$?
(a) $x = 0$ and $y=x + 6$
(b) $x = 0$ and $x=-2$
(c) $x = 0$ and $y=5x + 10$
(d) $x = 0$ and $y=x + 10$

Explanation:

Step1: Find vertical asymptotes

Set denominator $x^{2}+2x=x(x + 2)=0$. Solving gives $x = 0$ and $x=-2$. But canceling out common factors in numerator and denominator (if any), after factoring numerator as $(x + 1)(x + 2)(x + 5)$ and denominator as $x(x + 2)$, we cancel out $(x + 2)$. So vertical - asymptote is $x = 0$.

Step2: Determine slant asymptote

Since the degree of the numerator ($n = 3$) is one more than the degree of the denominator ($m=2$), we use polynomial long - division. Divide $x^{3}+8x^{2}+17x + 10$ by $x^{2}+2x$. The quotient is $x+6$. So the slant asymptote is $y=x + 6$. There is no horizontal asymptote as $n>m$.

Answer:

A. $x = 0$ and $y=x + 6$