Let $f(n)=n^3-5n^2+2n+8$ and $g(n)=n^3-6n^2+5n+12.$ What is the sum of all integers $n$ such that $\tfrac{f(n)}{g(n)}$ is an integer?
$$\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 5 \qquad \textbf{(E) } 6$$
Difficulty: 35/100 - A straightforward factorization problem which relies on avoiding the trick of extraneous solutions.
Core Concepts: polynomials, rational root theorem, factoring, extraneous solutions
Challenges: Competitors may be intimidated by the large polynomials and confused by the integer condition. Factorization of the cubic polynomials can also be somewhat difficult.
Techniques: Factoring the two cubics allows some terms to cancel out, allowing the contestant to find only two possibilities for $n$.
Error-prone Steps: Forgetting to discard the extraneous solution of $x=4$.
Ideal Time:
Experienced: ≤ 1-2 min
Intermediate: ≤ 3 min
Beginner: ≤ 6 min