(x^2-c)/(x-b)In the expression above, b and c are positive integers. If the expression is equivalent to x + b and...

1. TRANSLATE the problem information

• Given information:
  • Expression \(\frac{x^2-c}{x-b}\) is equivalent to \(x + b\)
  • \(x \neq b\), and \(b,c\) are positive integers
  • Need to find possible value of \(c\)

2. INFER the solution approach

• Since the expressions are equivalent, I can set up an equation
• The key insight: multiply both sides by \((x-b)\) to eliminate the denominator
• This will reveal the relationship between \(c\) and \(b\)

3. SIMPLIFY through algebraic steps

• Set up equation: \(\frac{x^2-c}{x-b} = x + b\)
• Multiply both sides by \((x-b)\): \(x^2 - c = (x + b)(x - b)\)
• Expand right side: \(x^2 - c = x^2 - b^2\)
• Subtract \(x^2\) from both sides: \(-c = -b^2\)
• Therefore: \(c = b^2\)

4. INFER the constraint on \(c\)

• Since \(c = b^2\) and \(b\) is a positive integer, \(c\) must be a perfect square
• Check which answer choice is a perfect square:
  • A. \(4 = 2^2\) ✓
  • B. 6 (not perfect square)
  • C. 8 (not perfect square)
  • D. 10 (not perfect square)

Answer: A. 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the strategic move to multiply both sides by \((x-b)\) to eliminate the denominator. Instead, they might try to factor the numerator directly or attempt polynomial long division without establishing the equivalence relationship first. Without this key insight, they can't discover that \(c = b^2\), leading to confusion about which answer choice to select. This leads to abandoning systematic solution and guessing.

Second Most Common Error:

Missing conceptual knowledge: Students may successfully work through the algebra to get \(c = b^2\) but don't recognize that this means \(c\) must be a perfect square. They might not immediately identify which of the answer choices are perfect squares, especially if they don't recall that \(4 = 2^2\). This conceptual gap may lead them to select Choice B (6) or another incorrect option based on other faulty reasoning.

The Bottom Line:

This problem requires recognizing both the strategic algebraic approach (multiply to eliminate denominators) and the conceptual insight about perfect squares. Students who miss either piece will struggle to reach the correct answer systematically.