Question:If x - 3 is a factor of the polynomial x^2 + bx - 15, where b is a constant,...

1. INFER the key relationship

• Given information:
  • \(\mathrm{(x - 3)}\) is a factor of \(\mathrm{x^2 + bx - 15}\)
  • We need to find the value of constant \(\mathrm{b}\)

• What this tells us: By the Factor Theorem, if \(\mathrm{(x - 3)}\) is a factor, then substituting \(\mathrm{x = 3}\) into the polynomial must give us zero.

2. SIMPLIFY by substituting \(\mathrm{x = 3}\)

• Set up the equation: \(\mathrm{(3)^2 + b(3) - 15 = 0}\)
• Calculate step by step:
  • \(\mathrm{(3)^2 = 9}\)
  • \(\mathrm{b(3) = 3b}\)
  • So: \(\mathrm{9 + 3b - 15 = 0}\)

3. SIMPLIFY to solve for \(\mathrm{b}\)

• Combine like terms: \(\mathrm{3b + 9 - 15 = 0}\)
• Simplify: \(\mathrm{3b - 6 = 0}\)
• Add 6 to both sides: \(\mathrm{3b = 6}\)
• Divide by 3: \(\mathrm{b = 2}\)

4. Verify the answer (optional but recommended)

• If \(\mathrm{b = 2}\), then our polynomial is \(\mathrm{x^2 + 2x - 15}\)
• Factor this: We need two numbers that multiply to \(\mathrm{-15}\) and add to \(\mathrm{2}\)
• Those numbers are \(\mathrm{5}\) and \(\mathrm{-3}\): \(\mathrm{(x + 5)(x - 3)}\)
• Confirmed: \(\mathrm{(x - 3)}\) is indeed a factor

Answer: C. 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not immediately recognize that they need to use the Factor Theorem. Instead, they might try to factor \(\mathrm{x^2 + bx - 15}\) directly without realizing they can use the given factor \(\mathrm{(x - 3)}\) to find \(\mathrm{b}\) first.

This leads to confusion because they can't factor a polynomial with an unknown coefficient, causing them to get stuck and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{(3)^2 + b(3) - 15 = 0}\) but make calculation errors, such as:

• Getting \(\mathrm{9 + 3b - 15 = -6 + 3b = 0}\), leading to \(\mathrm{b = 2}\) (this is actually correct)
• Or making sign errors: \(\mathrm{9 + 3b + 15 = 0}\), getting \(\mathrm{3b = -24}\), so \(\mathrm{b = -8}\)

This may lead them to select Choice A (-8) instead of the correct answer.

The Bottom Line:

This problem tests whether students can connect the concept of polynomial factors to the Factor Theorem. The key insight is recognizing that '\(\mathrm{x - 3}\) is a factor' gives you enough information to find the unknown coefficient by setting the polynomial equal to zero when \(\mathrm{x = 3}\).