The graph of a polynomial function f is shown in the xy-plane. The graph is a parabola opening upwards with...

1. TRANSLATE the problem information

• Given information:
  • Polynomial function \(\mathrm{f(x)}\) with x-intercepts at \(\mathrm{(-3, 0)}\) and \(\mathrm{(4, 0)}\)
  • The graph passes through \(\mathrm{(2, -10)}\)
• What this tells us: The values \(\mathrm{x = -3}\) and \(\mathrm{x = 4}\) make \(\mathrm{f(x) = 0}\), so they are roots of the polynomial

2. INFER the factorization using the Factor Theorem

• The Factor Theorem states: If c is a root of a polynomial, then \(\mathrm{(x - c)}\) is a factor
• Since \(\mathrm{x = -3}\) is a root: \(\mathrm{(x - (-3)) = (x + 3)}\) is a factor
• Since \(\mathrm{x = 4}\) is a root: \(\mathrm{(x - 4)}\) is a factor
• Therefore, both \(\mathrm{(x + 3)}\) and \(\mathrm{(x - 4)}\) must be factors of \(\mathrm{f(x)}\)

3. INFER which answer choice contains both required factors

• Check each option for the factors \(\mathrm{(x + 3)}\) and \(\mathrm{(x - 4)}\):
  • (A) \(\mathrm{(x - 4)(x - 2)}\): Has \(\mathrm{(x - 4)}\) ✓ but missing \(\mathrm{(x + 3)}\) ✗
  • (B) \(\mathrm{(x - 3)(x + 4)}\): Has neither required factor ✗
  • (C) \(\mathrm{(x + 3)(x - 4)}\): Has both required factors ✓
  • (D) \(\mathrm{(x + 3)(x - 2)}\): Has \(\mathrm{(x + 3)}\) ✓ but missing \(\mathrm{(x - 4)}\) ✗

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Making sign errors when converting from roots to factors

Students know that \(\mathrm{x = -3}\) is a root but incorrectly think this means \(\mathrm{(x - 3)}\) is a factor instead of \(\mathrm{(x + 3)}\). They forget that if \(\mathrm{x = -3}\) makes the polynomial zero, then \(\mathrm{(x - (-3)) = (x + 3)}\) is the factor. This sign confusion leads them to select Choice B: \(\mathrm{(x - 3)(x + 4)}\).

Second Most Common Error:

Poor TRANSLATE reasoning: Misunderstanding the significance of the point \(\mathrm{(2, -10)}\)

Students see that the graph passes through \(\mathrm{(2, -10)}\) and incorrectly assume this means \(\mathrm{x = 2}\) is also a root, forgetting that roots occur only where \(\mathrm{f(x) = 0}\), not where \(\mathrm{f(x) = -10}\). This leads them to favor answer choices containing \(\mathrm{(x - 2)}\) as a factor, causing them to select Choice A: \(\mathrm{(x - 4)(x - 2)}\) or Choice D: \(\mathrm{(x + 3)(x - 2)}\).

The Bottom Line:

Success on this problem requires clearly distinguishing between roots (where \(\mathrm{f(x) = 0}\)) and other points on the graph, plus careful attention to signs when applying the Factor Theorem.