Question:For real numbers x neq -2, the expression (2x^2 + 3x - 11)/(x + 2) can be written in the...

1. TRANSLATE the problem information

• Given: \(\mathrm{(2x^2 + 3x - 11)/(x + 2)}\) can be written as \(\mathrm{2x - 1 + k/(x + 2)}\)
• Need to find: The value of k

2. INFER the mathematical approach

• This is asking for the remainder when \(\mathrm{2x^2 + 3x - 11}\) is divided by \(\mathrm{x + 2}\)
• We can use either the Remainder Theorem OR polynomial long division
• Remainder Theorem is often faster: when dividing f(x) by (x + 2), the remainder equals f(-2)

3. SIMPLIFY using the Remainder Theorem

• Let \(\mathrm{f(x) = 2x^2 + 3x - 11}\)
• Find \(\mathrm{f(-2)}\): \(\mathrm{f(-2) = 2(-2)^2 + 3(-2) - 11}\)
• Calculate step by step:
  • \(\mathrm{2(-2)^2 = 2(4) = 8}\)
  • \(\mathrm{3(-2) = -6}\)
  • So: \(\mathrm{f(-2) = 8 - 6 - 11 = -9}\)

4. Verify with alternative method (polynomial long division)

• SIMPLIFY through division steps:
  • \(\mathrm{2x^2 \div x = 2x}\) → \(\mathrm{2x(x + 2) = 2x^2 + 4x}\)
  • Subtract: \(\mathrm{(2x^2 + 3x - 11) - (2x^2 + 4x) = -x - 11}\)
  • \(\mathrm{(-x) \div x = -1}\) → \(\mathrm{(-1)(x + 2) = -x - 2}\)
  • Subtract: \(\mathrm{(-x - 11) - (-x - 2) = -9}\)

Answer: B (-9)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing this as a remainder problem

Students may try to expand \(\mathrm{2x - 1 + k/(x + 2)}\) and set it equal to the original expression, leading to complex algebraic manipulation instead of using the straightforward remainder theorem or polynomial division. This leads to confusion and often abandoning the systematic approach for guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic errors when evaluating \(\mathrm{f(-2)}\)

Common mistakes include: \(\mathrm{2(-2)^2 = -8}\) instead of +8, or incorrect addition/subtraction of terms like \(\mathrm{8 - 6 - 11 = -13}\) instead of -9. These calculation errors may lead them to select Choice A (-12) or get confused when their answer doesn't match any choice.

The Bottom Line:

This problem tests whether students can recognize polynomial division structure and execute either the remainder theorem or long division accurately. The key insight is seeing that k represents the remainder, not trying to solve through complex algebraic manipulation.