The rational expression (12x^2 + mx - 28)/(3x + r) simplifies to 4x + s for all real x neq...

1. TRANSLATE the problem information

• Given information:
  • \(\frac{12\mathrm{x}^2 + \mathrm{mx} - 28}{3\mathrm{x} + \mathrm{r}}\) simplifies to \(4\mathrm{x} + \mathrm{s}\)
  • m, r, and s are all integers
  • This simplification works for all real \(\mathrm{x} \neq -\frac{\mathrm{r}}{3}\)

• What this tells us: The division has no remainder, so the numerator must equal the denominator times the quotient.

2. INFER the key relationship

• If the rational expression simplifies completely to \(4\mathrm{x} + \mathrm{s}\), then:
  \(12\mathrm{x}^2 + \mathrm{mx} - 28 = (3\mathrm{x} + \mathrm{r})(4\mathrm{x} + \mathrm{s})\)

• This polynomial equality must hold for all values of x.

3. SIMPLIFY by expanding the right side

• Expand \((3\mathrm{x} + \mathrm{r})(4\mathrm{x} + \mathrm{s})\):
  \((3\mathrm{x} + \mathrm{r})(4\mathrm{x} + \mathrm{s}) = 12\mathrm{x}^2 + 3\mathrm{sx} + 4\mathrm{rx} + \mathrm{rs}\)
  \(= 12\mathrm{x}^2 + (3\mathrm{s} + 4\mathrm{r})\mathrm{x} + \mathrm{rs}\)

4. INFER coefficient relationships

• Since \(12\mathrm{x}^2 + \mathrm{mx} - 28 = 12\mathrm{x}^2 + (3\mathrm{s} + 4\mathrm{r})\mathrm{x} + \mathrm{rs}\), the coefficients of like terms must be equal:
  • x² terms: \(12 = 12\) ✓
  • x terms: \(\mathrm{m} = 3\mathrm{s} + 4\mathrm{r}\)
  • constant terms: \(-28 = \mathrm{rs}\)

5. INFER which expression must be an integer

• From \(\mathrm{rs} = -28\) where r and s are integers:
  Since \(\mathrm{r} \times \mathrm{s} = -28\), r must be a divisor of -28

• Therefore:
  \(\frac{28}{\mathrm{r}} = \frac{28}{\mathrm{r}} \times \frac{-1}{-1}\)
  \(= \frac{-28}{\mathrm{r}} \times (-1)\)
  \(= (-\mathrm{s}) \times (-1)\)
  \(= \mathrm{s}\)

• Since s is an integer, \(\frac{28}{\mathrm{r}}\) must be an integer.

Answer: C (\(\frac{28}{\mathrm{r}}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that polynomial division simplification requires the numerator to equal denominator × quotient. Instead, they might try to work backwards from the answer choices or attempt polynomial long division without establishing the fundamental relationship.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Incomplete SIMPLIFY execution: Students correctly expand \((3\mathrm{x} + \mathrm{r})(4\mathrm{x} + \mathrm{s})\) but make arithmetic errors or fail to systematically match all coefficients. They might get \(\mathrm{m} = 3\mathrm{s} + 4\mathrm{r}\) but miss that \(\mathrm{rs} = -28\), preventing them from recognizing the divisibility relationship.

This may lead them to select Choice A (\(\frac{\mathrm{m}}{\mathrm{r}}\)) or Choice B (\(\frac{\mathrm{m}}{\mathrm{s}}\)) based on seeing these expressions in their work.

The Bottom Line:

This problem tests whether students can connect polynomial equality with coefficient matching, then apply divisibility concepts. The key insight is recognizing that \(\mathrm{rs} = -28\) with integer constraints forces a divisibility relationship that makes \(\frac{28}{\mathrm{r}}\) an integer.