\(\mathrm{g(x) = (x + 2)(x - 1)(x - 3)}\). The function g is defined above. Which of the following is...

1. INFER the approach needed

• Given: \(\mathrm{g(x) = (x + 2)(x - 1)(x - 3)}\) in factored form
• Need: To identify which point is NOT an x-intercept
• Strategy: Find all x-intercepts by setting \(\mathrm{g(x) = 0}\), then check which answer choice doesn't match

2. INFER how to find x-intercepts from factored form

• X-intercepts occur where \(\mathrm{g(x) = 0}\)
• Since \(\mathrm{g(x)}\) is already factored, set equal to zero:
  \(\mathrm{(x + 2)(x - 1)(x - 3) = 0}\)

3. INFER and apply the zero product property

• If a product equals zero, at least one factor must equal zero
• Set each factor equal to zero:
  • \(\mathrm{x + 2 = 0}\)
  • \(\mathrm{x - 1 = 0}\)
  • \(\mathrm{x - 3 = 0}\)

4. SIMPLIFY each equation

• \(\mathrm{x + 2 = 0}\) → \(\mathrm{x = -2}\)
• \(\mathrm{x - 1 = 0}\) → \(\mathrm{x = 1}\)
• \(\mathrm{x - 3 = 0}\) → \(\mathrm{x = 3}\)

5. INFER the x-intercept coordinates

• X-intercepts are points where \(\mathrm{y = 0}\)
• The x-intercepts are: \(\mathrm{(-2, 0)}\), \(\mathrm{(1, 0)}\), and \(\mathrm{(3, 0)}\)

6. INFER which answer choice doesn't fit

• Comparing to answer choices:
  • (A) \(\mathrm{(-2, 0)}\) ✓ matches our result
  • (B) \(\mathrm{(0, 0)}\) ✗ doesn't match any of our intercepts
  • (C) \(\mathrm{(1, 0)}\) ✓ matches our result
  • (D) \(\mathrm{(3, 0)}\) ✓ matches our result

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that the factored form gives direct access to the zeros through the zero product property.

Instead, they might try to expand the polynomial first: \(\mathrm{g(x) = (x + 2)(x - 1)(x - 3) = x^3 - 2x^2 - 5x + 6}\), then attempt to find roots of this cubic. This creates unnecessary complexity and increases chances of algebraic errors.

This leads to confusion and potentially guessing among the answer choices.

Second Most Common Error:

Conceptual confusion about x-intercepts: Students might think that since \(\mathrm{(0, 0)}\) is the origin, it's automatically an intercept, or they might confuse x-intercepts with y-intercepts.

They may not verify by substituting \(\mathrm{x = 0}\) into the function to check if \(\mathrm{g(0) = 0}\). This incomplete reasoning could lead them to incorrectly eliminate other choices and select an answer that IS actually an x-intercept.

The Bottom Line:

This problem rewards students who recognize the power of factored form - it makes finding zeros straightforward through the zero product property, rather than requiring complex algebraic manipulation.