If c is a constant, the polynomial 6x^2 - x - 15 is divisible by \(\mathrm{(2x + 3)}\). The result...

1. TRANSLATE the problem statement

• Given information:
  • Polynomial: \(6\mathrm{x}^2 - \mathrm{x} - 15\)
  • Divisible by: \((2\mathrm{x} + 3)\)
  • Quotient: \((3\mathrm{x} - \mathrm{c})\)
  • Need to find: value of c

• What this tells us: Division means dividend = divisor × quotient

2. TRANSLATE the relationship into an equation

• Since \(6\mathrm{x}^2 - \mathrm{x} - 15 \div (2\mathrm{x} + 3) = (3\mathrm{x} - \mathrm{c})\), we can write:

\((2\mathrm{x} + 3)(3\mathrm{x} - \mathrm{c}) = 6\mathrm{x}^2 - \mathrm{x} - 15\)

3. SIMPLIFY by expanding the left side

• Use FOIL method on \((2\mathrm{x} + 3)(3\mathrm{x} - \mathrm{c})\):
  • First terms: \((2\mathrm{x})(3\mathrm{x}) = 6\mathrm{x}^2\)
  • Outer terms: \((2\mathrm{x})(-\mathrm{c}) = -2\mathrm{cx}\)
  • Inner terms: \((3)(3\mathrm{x}) = 9\mathrm{x}\)
  • Last terms: \((3)(-\mathrm{c}) = -3\mathrm{c}\)

• Combine: \(6\mathrm{x}^2 - 2\mathrm{cx} + 9\mathrm{x} - 3\mathrm{c} = 6\mathrm{x}^2 + (9 - 2\mathrm{c})\mathrm{x} - 3\mathrm{c}\)

4. INFER that coefficients must be equal

• Since \(6\mathrm{x}^2 + (9 - 2\mathrm{c})\mathrm{x} - 3\mathrm{c} = 6\mathrm{x}^2 - \mathrm{x} - 15\), the coefficients of like terms must match:
  • x² coefficient: \(6 = 6\) ✓
  • x coefficient: \(9 - 2\mathrm{c} = -1\)
  • Constant term: \(-3\mathrm{c} = -15\)

5. SIMPLIFY to solve for c

• From constant terms: \(-3\mathrm{c} = -15\)
• Divide both sides by -3: \(\mathrm{c} = 5\)
• Verify with x terms: \(9 - 2(5) = 9 - 10 = -1\) ✓

Answer: D) 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that "divisible by \((2\mathrm{x} + 3)\) with quotient \((3\mathrm{x} - \mathrm{c})\)" means \((2\mathrm{x} + 3)(3\mathrm{x} - \mathrm{c})\) equals the original polynomial. Instead, they might try to perform long division directly or set up incorrect equations like \((2\mathrm{x} + 3) + (3\mathrm{x} - \mathrm{c}) = 6\mathrm{x}^2 - \mathrm{x} - 15\).

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \((2\mathrm{x} + 3)(3\mathrm{x} - \mathrm{c}) = 6\mathrm{x}^2 - \mathrm{x} - 15\) but make algebraic errors when expanding, such as:

• Getting signs wrong: writing +2cx instead of -2cx
• Forgetting terms when using FOIL
• Making errors when combining like terms

This typically leads them to select Choice A (-5) or Choice B (1) based on their incorrect expansion.

The Bottom Line:

This problem requires students to translate a division concept into multiplication form and then execute careful algebraic manipulation. Students who rush through the FOIL expansion or don't fully understand the division-multiplication relationship will struggle to find the correct parameter value.