College Algebra, Chapter 4, Chapter Review, Section Review, Problem 76

Analyze the graph of the rational function $\displaystyle r(x) = \frac{2x^3 - x^2}{x + 1}$ by using a graphing device. Find all $x$ and $y$ intercepts and all vertical, horizontal and slant asymptotes. If the function has no horizontal or slant asymptote, find a polynomial that has the same end as the rational function.







Based from the graph, the $x$ intercepts are and $0.5$ On the other hand, the value of $y$ intercept is . Also, the vertical asymptote is $x = -1$. More over, since the degree of the numerator is greater than the degree in the denominator by a factor of $2$, then the rational function has no horizontal or slant asymptote. Thus by applying long division,







Hence, if $\displaystyle r(x) = \frac{2x^3 - x^2}{x + 1} = 2x^2 - 3x + 3 - \frac{3}{x + 1}$

Then, the polynomial $f(x) = 2x^2 - 3x + 3$ has the same end behavior as the given rational function.