Question:For a polynomial \(\mathrm{P(x) = x^3 + kx + 18c}\), where k is a constant and c is a positive...

1. INFER the key relationship using the Factor Theorem

• Given information:
  • \(\mathrm{P(x) = x^3 + kx + 18c}\)
  • \(\mathrm{(x - 3c)}\) is a factor of \(\mathrm{P(x)}\)
  • \(\mathrm{c}\) is a positive integer
• What this tells us: By the Factor Theorem, if \(\mathrm{(x - 3c)}\) is a factor, then \(\mathrm{P(3c) = 0}\)

2. TRANSLATE this relationship into an equation

• Substitute \(\mathrm{x = 3c}\) into \(\mathrm{P(x)}\):
  \(\mathrm{P(3c) = (3c)^3 + k(3c) + 18c = 0}\)
• This gives us: \(\mathrm{27c^3 + 3kc + 18c = 0}\)

3. SIMPLIFY the equation to find the relationship between k and c

• Factor out 3c: \(\mathrm{3c(9c^2 + k + 6) = 0}\)
• Since \(\mathrm{c \gt 0}\), we know \(\mathrm{3c \neq 0}\), so: \(\mathrm{9c^2 + k + 6 = 0}\)
• Solve for \(\mathrm{c^2}\): \(\mathrm{c^2 = (-k - 6)/9}\)

4. APPLY CONSTRAINTS to determine valid values

• For \(\mathrm{c}\) to be a positive integer, \(\mathrm{(-k - 6)/9}\) must be a positive perfect square
• Test each choice:
  • Choice A: \(\mathrm{k = -78}\) → \(\mathrm{c^2 = 72/9 = 8}\) → \(\mathrm{c = 2\sqrt{2}}\) (not an integer)
  • Choice B: \(\mathrm{k = -42}\) → \(\mathrm{c^2 = 36/9 = 4}\) → \(\mathrm{c = 2}\) ✓
  • Choice C: \(\mathrm{k = -22}\) → \(\mathrm{c^2 = 16/9}\) → \(\mathrm{c = 4/3}\) (not an integer)
  • Choice D: \(\mathrm{k = -10}\) → \(\mathrm{c^2 = 4/9}\) → \(\mathrm{c = 2/3}\) (not an integer)

Answer: B (-42)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize that they should apply the Factor Theorem immediately. Instead, they might try to factor the polynomial directly or use polynomial division, which becomes unnecessarily complex with the unknown constants.

Without the Factor Theorem insight, students get bogged down in algebraic manipulation that doesn't lead to a clear path forward. This leads to confusion and guessing.

Second Most Common Error:

Poor APPLY CONSTRAINTS execution: Students correctly set up \(\mathrm{P(3c) = 0}\) and solve to get \(\mathrm{c^2 = (-k - 6)/9}\), but fail to recognize that \(\mathrm{c}\) must be a positive integer. They might select the first choice where \(\mathrm{(-k - 6)/9}\) is positive, without checking whether it yields an integer value for c.

This may lead them to select Choice A (-78) since it gives a positive value under the square root, even though \(\mathrm{c = 2\sqrt{2}}\) is not an integer.

The Bottom Line:

This problem tests whether students can connect the abstract concept of polynomial factors to the concrete Factor Theorem, then systematically apply constraints to eliminate impossible answers. The key insight is that "factor" immediately suggests "Factor Theorem," not complex algebraic manipulation.