Suppose $a$ and $b$ are real numbers. When the polynomial $x^3+x^2+ax+b$ is divided by $x-1$, the remainder is $4$. When divided by $x-2$, the remainder is $6$. What is $b-a$?
$$\textbf{(A) } 14 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 18$$
Difficulty: 18/100 — Requires remainder theorem and solving a linear system. Fairly straightforward, but tedious without the theorem.
Core Concepts: polynomials, remainder theorem
Techniques: Plugging and chugging once the theorem is directly applied.
Error-prone Steps: Mixing up linear expressions, arithmetic errors, extracting the incorrect result.
Ideal Time:
Experienced: ≤ 30 sec
Intermediate: ≤ 40 sec
Beginner: ≤ 1-1.5 min