The polynomial 6x^3 + mx^2 + nx - 10 can be written as \((3\mathrm{x}^2 + \mathrm{k}\mathrm{x} - 5)(\mathrm{a}\mathrm{x} + \mathrm{b})\),...

1. TRANSLATE the problem setup

• Given information:
  • Original polynomial: \(6\mathrm{x}^3 + \mathrm{m}\mathrm{x}^2 + \mathrm{n}\mathrm{x} - 10\)
  • Factored form: \((3\mathrm{x}^2 + \mathrm{k}\mathrm{x} - 5)(\mathrm{a}\mathrm{x} + \mathrm{b})\)
  • k, a, and b are integers
  • Need to find which expression must always be an integer

2. INFER the solution strategy

• Key insight: If these two expressions are equal, their coefficients must match
• Strategy: Expand the factored form, then compare coefficients to find relationships
• This will let us express m and n in terms of k, a, and b

3. SIMPLIFY the factored form expansion

Expand \((3\mathrm{x}^2 + \mathrm{k}\mathrm{x} - 5)(\mathrm{a}\mathrm{x} + \mathrm{b})\):

• First term × both: \(3\mathrm{a}\mathrm{x}^3 + 3\mathrm{b}\mathrm{x}^2\)
• Second term × both: \(\mathrm{a}\mathrm{k}\mathrm{x}^2 + \mathrm{b}\mathrm{k}\mathrm{x}\)
• Third term × both: \(-5\mathrm{a}\mathrm{x} - 5\mathrm{b}\)
• Combine like terms: \(3\mathrm{a}\mathrm{x}^3 + (3\mathrm{b} + \mathrm{a}\mathrm{k})\mathrm{x}^2 + (\mathrm{b}\mathrm{k} - 5\mathrm{a})\mathrm{x} - 5\mathrm{b}\)

4. INFER coefficient relationships

Compare \(6\mathrm{x}^3 + \mathrm{m}\mathrm{x}^2 + \mathrm{n}\mathrm{x} - 10 = 3\mathrm{a}\mathrm{x}^3 + (3\mathrm{b} + \mathrm{a}\mathrm{k})\mathrm{x}^2 + (\mathrm{b}\mathrm{k} - 5\mathrm{a})\mathrm{x} - 5\mathrm{b}\):

• x³ coefficient: \(3\mathrm{a} = 6\) → \(\mathrm{a} = 2\)
• Constant term: \(-5\mathrm{b} = -10\) → \(\mathrm{b} = 2\)
• x² coefficient: \(\mathrm{m} = 3\mathrm{b} + \mathrm{a}\mathrm{k} = 3(2) + 2\mathrm{k} = 6 + 2\mathrm{k}\)
• x¹ coefficient: \(\mathrm{n} = \mathrm{b}\mathrm{k} - 5\mathrm{a} = 2\mathrm{k} - 5(2) = 2\mathrm{k} - 10\)

5. SIMPLIFY each answer choice

Now substitute our expressions for m and n:

(A) \(\frac{\mathrm{m}}{\mathrm{k}}\)

\(\frac{\mathrm{m}}{\mathrm{k}} = \frac{6 + 2\mathrm{k}}{\mathrm{k}}\)
\(= \frac{6}{\mathrm{k}} + 2\)

This is an integer only when k divides 6 evenly

(B) \(\frac{\mathrm{n}}{\mathrm{k}}\)

\(\frac{\mathrm{n}}{\mathrm{k}} = \frac{2\mathrm{k} - 10}{\mathrm{k}}\)
\(= 2 - \frac{10}{\mathrm{k}}\)

This is an integer only when k divides 10 evenly

(C) \(\frac{\mathrm{m}+\mathrm{n}}{2}\)

\(\frac{\mathrm{m}+\mathrm{n}}{2} = \frac{6 + 2\mathrm{k} + 2\mathrm{k} - 10}{2}\)
\(= \frac{4\mathrm{k} - 4}{2}\)
\(= 2\mathrm{k} - 2\)

Since k is an integer, 2k - 2 is always an integer

(D) \(\frac{\mathrm{m}-\mathrm{n}}{6}\)

\(\frac{\mathrm{m}-\mathrm{n}}{6} = \frac{6 + 2\mathrm{k} - (2\mathrm{k} - 10)}{6}\)
\(= \frac{16}{6}\)
\(= \frac{8}{3}\)

This is never an integer

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students often struggle to recognize that coefficient comparison is the key strategy. They might try to factor the original polynomial directly or get overwhelmed by the multiple variables (m, n, k, a, b).

Without this strategic insight, they may attempt algebraic manipulation without direction, leading to confusion and random guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the strategy but make algebraic errors when expanding or comparing coefficients. A common mistake is incorrectly handling the signs when expanding \((3\mathrm{x}^2 + \mathrm{k}\mathrm{x} - 5)(\mathrm{a}\mathrm{x} + \mathrm{b})\), especially with the \(-5\mathrm{a}\mathrm{x}\) term.

This leads to wrong expressions for m and n, causing them to select Choice A or B based on incorrect calculations.

The Bottom Line:

This problem tests whether students can systematically use polynomial equality (same coefficients) to create a system of relationships. The key insight is recognizing that only expressions involving integer operations on integer variables will always yield integers.