The function f is defined by \(\mathrm{f(x) = (x - 7)(x + 4)(x + 3)}\). Which of the following is...

1. INFER what x-intercepts mean mathematically

• Given: \(\mathrm{f(x) = (x - 7)(x + 4)(x + 3)}\)
• Key insight: X-intercepts occur where the graph crosses the x-axis
• This means we need points where \(\mathrm{f(x) = 0}\)

2. INFER the solution strategy using zero product property

• Since \(\mathrm{f(x)}\) is already factored as a product: \(\mathrm{(x - 7)(x + 4)(x + 3)}\)
• When this product equals zero, at least one factor must equal zero
• This gives us three separate equations to solve

3. SIMPLIFY by solving each factor equation

• From \(\mathrm{x - 7 = 0}\): \(\mathrm{x = 7}\)
• From \(\mathrm{x + 4 = 0}\): \(\mathrm{x = -4}\)
• From \(\mathrm{x + 3 = 0}\): \(\mathrm{x = -3}\)

4. Match solutions to answer choices

• Our x-intercepts: 7, -4, -3
• Available choices: (A) -7, (B) 3, (C) 4, (D) 7
• Only \(\mathrm{x = 7}\) appears in the choices

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about x-intercepts: Students may not connect that "x-intercept" means "where \(\mathrm{f(x) = 0}\)"

Instead of setting the function equal to zero, they might try to substitute the answer choices into the original function or look for patterns in the coefficients. This leads to confusion and guessing among the choices.

Second Most Common Error:

Poor SIMPLIFY execution with signs: When solving \(\mathrm{x + 4 = 0}\) or \(\mathrm{x + 3 = 0}\), students make sign errors

For example, solving \(\mathrm{x + 4 = 0}\) incorrectly as \(\mathrm{x = 4}\) (instead of \(\mathrm{x = -4}\)). Since 4 appears as choice (C), this computational error could lead them to select Choice C (4).

The Bottom Line:

This problem tests whether students understand that x-intercepts are solutions to \(\mathrm{f(x) = 0}\), combined with proper application of the zero product property. The factored form makes the algebra simple—the challenge is recognizing the right approach and executing it carefully.