One of the factors of x^3 + 10x^2 + 19x - 30 is x + b, where b is a...

1. TRANSLATE the problem information

• Given information:
  • Polynomial: \(\mathrm{x^3 + 10x^2 + 19x - 30}\)
  • One factor has the form \(\mathrm{x + b}\) where \(\mathrm{b}\) is positive
  • Need to find the smallest possible value of \(\mathrm{b}\)
• What this tells us: We need to find the roots of the polynomial and identify which give us factors with positive \(\mathrm{b}\).

2. INFER the approach strategy

• Key insight: If \(\mathrm{x + b}\) is a factor with positive \(\mathrm{b}\), then \(\mathrm{x = -b}\) must be a root (where \(\mathrm{b \gt 0}\))
• This means we should test negative values when looking for roots
• Strategy: Use Rational Root Theorem to find possible roots, then test systematically

3. APPLY Rational Root Theorem to find candidates

• Possible rational roots = ±(factors of constant term)/(factors of leading coefficient)
• Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
• Possible roots: \(\mathrm{±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30}\)

4. SIMPLIFY by testing negative candidates (since we want positive b)

Testing \(\mathrm{x = -1}\):

\(\mathrm{(-1)^3 + 10(-1)^2 + 19(-1) - 30}\)
\(\mathrm{= -1 + 10 - 19 - 30}\)
\(\mathrm{= -40 ≠ 0}\)

Testing \(\mathrm{x = -5}\):

\(\mathrm{(-5)^3 + 10(-5)^2 + 19(-5) - 30}\)
\(\mathrm{= -125 + 250 - 95 - 30}\)
\(\mathrm{= 0}\) ✓

So \(\mathrm{x = -5}\) is a root, meaning \(\mathrm{(x + 5)}\) is a factor with \(\mathrm{b = 5}\).

Testing \(\mathrm{x = -6}\):

\(\mathrm{(-6)^3 + 10(-6)^2 + 19(-6) - 30}\)
\(\mathrm{= -216 + 360 - 114 - 30}\)
\(\mathrm{= 0}\) ✓

So \(\mathrm{x = -6}\) is a root, meaning \(\mathrm{(x + 6)}\) is a factor with \(\mathrm{b = 6}\).

5. APPLY CONSTRAINTS to select final answer

• We found factors \(\mathrm{(x + 5)}\) and \(\mathrm{(x + 6)}\) with positive \(\mathrm{b}\) values of 5 and 6
• Since we need the smallest possible value: \(\mathrm{b = 5}\)

Answer: B) 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the connection between wanting \(\mathrm{x + b}\) (with positive \(\mathrm{b}\)) and needing to test negative root values.

Instead, they might test positive values like \(\mathrm{x = 1, 2, 3}\), etc., looking for factors of the form \(\mathrm{(x - a)}\). When these don't work out cleanly or they find \(\mathrm{x = 1}\) as a root (giving factor \(\mathrm{x - 1}\)), they get confused about how to get a factor \(\mathrm{x + b}\) with positive \(\mathrm{b}\). This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when evaluating the polynomial at test values, particularly with the mix of positive and negative terms and higher powers.

For example, when testing \(\mathrm{x = -5}\), they might calculate \(\mathrm{(-5)^3}\) as \(\mathrm{+125}\) instead of \(\mathrm{-125}\), or make sign errors in combining terms like \(\mathrm{-125 + 250 - 95 - 30}\). These calculation errors cause them to miss actual roots or think non-roots are valid. This may lead them to select Choice A (1) if they incorrectly think there are no factors with positive \(\mathrm{b}\) and guess the smallest answer.

The Bottom Line:

This problem requires strategic thinking about the relationship between factor forms and root signs. Students who approach it mechanically without understanding why negative test values are needed for positive \(\mathrm{b}\) factors will struggle to find the systematic path to the solution.