The expression 15x^2 + bx - 20, where b is a constant, can be rewritten as \((\mathrm{hx} + \mathrm{k})(\mathrm{jx} +...

1. TRANSLATE the problem information

• Given: \(15\mathrm{x}^2 + \mathrm{bx} - 20\) can be written as \((\mathrm{hx} + \mathrm{k})(\mathrm{jx} + \mathrm{m})\)
• All of h, j, k, m, b are integer constants
• Need to find which expression must always be an integer

2. INFER the strategy needed

• To connect the two forms, I need to expand the factored form
• Then compare coefficients to find relationships between the constants
• Use these relationships to analyze each answer choice

3. SIMPLIFY by expanding the factored form

Expanding \((\mathrm{hx} + \mathrm{k})(\mathrm{jx} + \mathrm{m})\):

\((\mathrm{hx} + \mathrm{k})(\mathrm{jx} + \mathrm{m}) = \mathrm{hjx}^2 + \mathrm{hmx} + \mathrm{jkx} + \mathrm{km}\)

\(= \mathrm{hjx}^2 + (\mathrm{hm} + \mathrm{jk})\mathrm{x} + \mathrm{km}\)

4. INFER coefficient relationships

Comparing \(\mathrm{hjx}^2 + (\mathrm{hm} + \mathrm{jk})\mathrm{x} + \mathrm{km}\) with \(15\mathrm{x}^2 + \mathrm{bx} - 20\):

• Coefficient of \(\mathrm{x}^2\): \(\mathrm{hj} = 15\)
• Coefficient of x: \(\mathrm{hm} + \mathrm{jk} = \mathrm{b}\)
• Constant term: \(\mathrm{km} = -20\)

5. CONSIDER ALL CASES by analyzing each answer choice

For choice (D): Since \(\mathrm{km} = -20\) and both k and m are integers, k must be a divisor of -20. The divisors of -20 are: ±1, ±2, ±4, ±5, ±10, ±20. Since k divides 20, the expression \(20/\mathrm{k}\) must always be an integer.

For the other choices, let me check if they can be non-integers using a counterexample:

Let \(\mathrm{h} = 3\), \(\mathrm{j} = 5\), \(\mathrm{k} = 5\), \(\mathrm{m} = -4\)

Verify this works:

\(\mathrm{hj} = 15\) ✓

\(\mathrm{km} = -20\) ✓

\(\mathrm{b} = \mathrm{hm} + \mathrm{jk}\)

\(= 3(-4) + 5(5)\)

\(= 13\)

• (A) \(\mathrm{b/h} = 13/3 \approx 4.33\) (not an integer)
• (B) \(\mathrm{b/k} = 13/5 = 2.6\) (not an integer)
• (C) \(20/\mathrm{h} = 20/3 \approx 6.67\) (not an integer)
• (D) \(20/\mathrm{k} = 20/5 = 4\) (integer ✓)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students correctly expand the polynomial but fail to recognize the key insight about divisibility. They see that \(\mathrm{km} = -20\) but don't connect this to the fact that k must divide -20, which guarantees that \(20/\mathrm{k}\) is an integer.

Instead, they might try to construct specific examples or get overwhelmed trying to analyze all the relationships simultaneously. This leads to confusion and guessing between the answer choices.

Second Most Common Error:

Incomplete CONSIDER ALL CASES execution: Students may correctly identify that \(20/\mathrm{k}\) must be an integer but fail to verify that the other options can be non-integers. Without testing counterexamples, they might incorrectly assume multiple answers could work.

This may lead them to select Choice A or Choice B, thinking that since b involves both h and k, ratios like \(\mathrm{b/h}\) might also have special integer properties.

The Bottom Line:

This problem tests whether students can connect polynomial coefficient relationships to integer divisibility properties. The key insight is recognizing that \(\mathrm{km} = -20\) with integer k forces a divisibility condition, making this a number theory problem disguised as a polynomial problem.