One of the factors of 2x^3 + 42x^2 + 208x is x + b, where b is a positive constant....

1. TRANSLATE the problem information

• Given information:
  • Expression: \(2\mathrm{x}^3 + 42\mathrm{x}^2 + 208\mathrm{x}\)
  • One factor has form \(\mathrm{x} + \mathrm{b}\) where \(\mathrm{b}\) is positive
• Need to find: smallest possible value of \(\mathrm{b}\)

2. INFER the solution approach

• To find all factors of the form \(\mathrm{x} + \mathrm{b}\), I need to factor the polynomial completely
• Start by looking for a greatest common factor

3. SIMPLIFY by factoring out the GCF

• Each term has factors of 2 and \(\mathrm{x}\):

\(2\mathrm{x}^3 + 42\mathrm{x}^2 + 208\mathrm{x} = 2\mathrm{x}(\mathrm{x}^2 + 21\mathrm{x} + 104)\)

4. SIMPLIFY the remaining quadratic factor

• Need to factor \(\mathrm{x}^2 + 21\mathrm{x} + 104\)
• Looking for two numbers that multiply to 104 and add to 21
• Test factor pairs of 104:
  • \(1 \times 104 = 104\), sum = 105 ✗
  • \(2 \times 52 = 104\), sum = 54 ✗
  • \(4 \times 26 = 104\), sum = 30 ✗
  • \(8 \times 13 = 104\), sum = 21 ✓
• Therefore: \(\mathrm{x}^2 + 21\mathrm{x} + 104 = (\mathrm{x} + 8)(\mathrm{x} + 13)\)

5. INFER all factors and identify candidates

• Complete factorization: \(2\mathrm{x}^3 + 42\mathrm{x}^2 + 208\mathrm{x} = 2\mathrm{x}(\mathrm{x} + 8)(\mathrm{x} + 13)\)
• Factors of the form \(\mathrm{x} + \mathrm{b}\) where \(\mathrm{b}\) is positive:
  • \(\mathrm{x} + 8\) (so \(\mathrm{b} = 8\))
  • \(\mathrm{x} + 13\) (so \(\mathrm{b} = 13\))

6. APPLY CONSTRAINTS to select final answer

• Since we need the smallest possible value of \(\mathrm{b}\):
• Between \(\mathrm{b} = 8\) and \(\mathrm{b} = 13\), the smallest is \(\mathrm{b} = 8\)

Answer: 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students may not factor the polynomial completely, stopping after factoring out \(2\mathrm{x}\).

They might think the only factors are \(2\mathrm{x}\) and \(\mathrm{x}^2 + 21\mathrm{x} + 104\), missing that the quadratic factors further. Since \(\mathrm{x}^2 + 21\mathrm{x} + 104\) is not in the form \(\mathrm{x} + \mathrm{b}\), they get stuck and may guess randomly or incorrectly conclude there are no factors of the required form.

Second Most Common Error:

Poor INFER reasoning: Students correctly factor to get \(2\mathrm{x}(\mathrm{x} + 8)(\mathrm{x} + 13)\) but fail to identify which factors have the form \(\mathrm{x} + \mathrm{b}\).

They might incorrectly consider compound factors like \(2\mathrm{x}\), \(2(\mathrm{x} + 8)\), or \(\mathrm{x}(\mathrm{x} + 8)\) as candidates, leading to confusion about what constitutes a "factor of the form \(\mathrm{x} + \mathrm{b}\)" and potentially selecting an incorrect answer.

The Bottom Line:

This problem requires systematic polynomial factorization skills combined with the ability to identify which factors match a specific algebraic form. Students must complete the entire factorization process, not just the first step.