If k - x is a factor of the expression -{x^2 + 1/29nk^2}, where n and k are constants and...

1. INFER the key relationship

• Given: \(\mathrm{(k - x)}\) is a factor of \(\mathrm{-x^2 + \frac{1}{29}nk^2}\)
• Key insight: If \(\mathrm{(k - x)}\) is a factor, then substituting \(\mathrm{x = k}\) should make the entire expression equal to 0
• This comes from the factor theorem

2. TRANSLATE and substitute

• Substitute \(\mathrm{x = k}\) into the expression:
  \(\mathrm{-k^2 + \frac{1}{29}nk^2}\)

3. SIMPLIFY to find n

• Set the expression equal to 0:
  \(\mathrm{-k^2 + \frac{1}{29}nk^2 = 0}\)
• Factor out \(\mathrm{k^2}\):
  \(\mathrm{k^2(-1 + \frac{1}{29}n) = 0}\)
• Since \(\mathrm{k \gt 0}\), we know \(\mathrm{k^2 ≠ 0}\), so:
  \(\mathrm{-1 + \frac{1}{29}n = 0}\)
• Add 1 to both sides:
  \(\mathrm{\frac{1}{29}n = 1}\)
• Multiply both sides by 29:
  \(\mathrm{n = 29}\)

4. INFER verification (optional but helpful)

• With \(\mathrm{n = 29}\), our expression becomes: \(\mathrm{-x^2 + k^2}\)
• This is equivalent to \(\mathrm{k^2 - x^2}\), which factors as \(\mathrm{(k - x)(k + x)}\)
• Indeed, \(\mathrm{(k - x)}\) is a factor! ✓

Answer: D. 29

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the connection between "factor" and what that means algebraically. Students may see the word "factor" but not understand that this means the expression can be written as \(\mathrm{(k - x)}\) times something else, or that they can use the factor theorem to substitute \(\mathrm{x = k}\).

This leads to confusion about how to even start the problem, causing them to guess randomly among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students might correctly set up \(\mathrm{-k^2 + \frac{1}{29}nk^2 = 0}\) but then make algebraic errors when solving for n. For example, they might forget to factor out \(\mathrm{k^2}\) properly or make sign errors when isolating n.

This could lead them to select incorrect values like Choice A (-29) or Choice B (-1/29).

The Bottom Line:

This problem tests whether students can bridge the gap between the conceptual meaning of "factor" and its practical algebraic implications. The key breakthrough is recognizing that factors give you information about when expressions equal zero.