When polynomial x^2 + ax + b is divided by \(\mathrm{(x - 1)}\), the quotient is x + 4 and...

1. TRANSLATE the division relationship

• Given information:
  • Dividend: \(\mathrm{x^2 + ax + b}\)
  • Divisor: \(\mathrm{(x - 1)}\)
  • Quotient: \(\mathrm{x + 4}\)
  • Remainder: \(\mathrm{6}\)

• We need to find the values of \(\mathrm{a}\) and \(\mathrm{b}\), then calculate \(\mathrm{a + b}\)

2. INFER the algebraic approach

• The division algorithm for polynomials states: \(\mathrm{dividend = divisor × quotient + remainder}\)
• This gives us: \(\mathrm{x^2 + ax + b = (x - 1)(x + 4) + 6}\)
• We can expand the right side and compare coefficients to find \(\mathrm{a}\) and \(\mathrm{b}\)

3. SIMPLIFY by expanding the right side

• First expand \(\mathrm{(x - 1)(x + 4)}\):
  \(\mathrm{(x - 1)(x + 4) = x^2 + 4x - x - 4}\)
  \(\mathrm{= x^2 + 3x - 4}\)

• Now add the remainder:
  \(\mathrm{(x - 1)(x + 4) + 6 = x^2 + 3x - 4 + 6}\)
  \(\mathrm{= x^2 + 3x + 2}\)

4. INFER by comparing coefficients

• We now have: \(\mathrm{x^2 + ax + b = x^2 + 3x + 2}\)
• Comparing coefficients of like terms:
  • \(\mathrm{x^2}\) terms: \(\mathrm{1 = 1}\) ✓
  • \(\mathrm{x}\) terms: \(\mathrm{a = 3}\)
  • Constant terms: \(\mathrm{b = 2}\)

5. Calculate the final answer

• \(\mathrm{a + b = 3 + 2 = 5}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Not knowing or forgetting the division algorithm for polynomials

Students may try to substitute specific values of \(\mathrm{x}\) or attempt to divide polynomials directly without using the fundamental relationship. Without the division algorithm, they can't set up the crucial equation \(\mathrm{x^2 + ax + b = (x - 1)(x + 4) + 6}\), making the problem unsolvable through systematic methods.

This leads to confusion and guessing.

Second Most Common Error:

Weak SIMPLIFY execution: Making algebraic errors when expanding \(\mathrm{(x - 1)(x + 4)}\)

Students might incorrectly expand to get \(\mathrm{x^2 + 3x + 4}\) or \(\mathrm{x^2 + 5x - 4}\), leading to wrong values for \(\mathrm{a}\) and \(\mathrm{b}\). For example, if they get \(\mathrm{x^2 + 5x - 4 + 6 = x^2 + 5x + 2}\), they would find \(\mathrm{a = 5}\) and \(\mathrm{b = 2}\), giving \(\mathrm{a + b = 7}\).

Since \(\mathrm{7}\) is not among the answer choices, this leads to confusion and random selection.

The Bottom Line:

This problem tests whether students can connect the conceptual understanding of polynomial division with algebraic manipulation. Success requires both knowing the division algorithm and executing the algebra correctly.