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Find the oblique (slant) asymptote of \(\displaystyle f(x) = \frac{x^{3} + 7 x^{2} - 80 x - 576}{x^{2} + 14 x + 45}\).
The oblique asymptote is the whole part without the remainder after polynomial long division.
\begin{question}Find the oblique (slant) asymptote of $f(x) = \frac{x^{3} + 7 x^{2} - 80 x - 576}{x^{2} + 14 x + 45}$.
\soln{9cm}{The oblique asymptote is the whole part without the remainder after polynomial long division. \newline
\begin{center}\begin{tikzpicture}
\draw (0,0) node{$ \polylongdiv{x^3 + 7*x^2 - 80*x - 576}{x^2 + 14*x + 45} $};
\end{tikzpicture}
\end{center}
Dividing gives the quotient $x - 7$ and the remainder $- 27 x - 261$. The oblique asymptote is $y = x - 7$. }
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Find the oblique (slant) asymptote of <img class="equation_image" title=" \displaystyle f(x) = \frac{x^{3} + 7 x^{2} - 80 x - 576}{x^{2} + 14 x + 45} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20%5Cfrac%7Bx%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20-%2080%20x%20-%20576%7D%7Bx%5E%7B2%7D%20%2B%2014%20x%20%2B%2045%7D%20" alt="LaTeX: \displaystyle f(x) = \frac{x^{3} + 7 x^{2} - 80 x - 576}{x^{2} + 14 x + 45} " data-equation-content=" \displaystyle f(x) = \frac{x^{3} + 7 x^{2} - 80 x - 576}{x^{2} + 14 x + 45} " /> . </p> </p><p> <p>The oblique asymptote is the whole part without the remainder after polynomial long division. <br>
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Dividing gives the quotient <img class="equation_image" title=" \displaystyle x - 7 " src="/equation_images/%20%5Cdisplaystyle%20x%20-%207%20" alt="LaTeX: \displaystyle x - 7 " data-equation-content=" \displaystyle x - 7 " /> and the remainder <img class="equation_image" title=" \displaystyle - 27 x - 261 " src="/equation_images/%20%5Cdisplaystyle%20-%2027%20x%20-%20261%20" alt="LaTeX: \displaystyle - 27 x - 261 " data-equation-content=" \displaystyle - 27 x - 261 " /> . The oblique asymptote is <img class="equation_image" title=" \displaystyle y = x - 7 " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20x%20-%207%20" alt="LaTeX: \displaystyle y = x - 7 " data-equation-content=" \displaystyle y = x - 7 " /> . </p> </p>