\(\mathrm{f(x) = (x + 4)(x - 1)(2x - 3)}\). The function f is defined above. Which of the following is...

1. TRANSLATE the problem information

• Given: \(\mathrm{f(x) = (x + 4)(x - 1)(2x - 3)}\)
• Find: Which point is NOT an x-intercept
• Key insight: X-intercepts occur where the graph crosses the x-axis (where \(\mathrm{y = 0}\))

2. INFER the approach

• To find x-intercepts, set \(\mathrm{f(x) = 0}\)
• Since we have a product of factors, use the zero product property
• Each factor that equals zero gives us an x-intercept

3. SIMPLIFY by setting up the equation

Set \(\mathrm{f(x) = 0}\):

\(\mathrm{0 = (x + 4)(x - 1)(2x - 3)}\)

4. INFER using zero product property

If a product equals zero, at least one factor must equal zero:

• \(\mathrm{x + 4 = 0}\) OR \(\mathrm{x - 1 = 0}\) OR \(\mathrm{2x - 3 = 0}\)

5. SIMPLIFY each linear equation

• From \(\mathrm{x + 4 = 0}\): \(\mathrm{x = -4}\)
• From \(\mathrm{x - 1 = 0}\): \(\mathrm{x = 1}\)
• From \(\mathrm{2x - 3 = 0}\): \(\mathrm{x = \frac{3}{2}}\)

6. TRANSLATE back to coordinate points

The x-intercepts are: \(\mathrm{(-4, 0)}\), \(\mathrm{(1, 0)}\), and \(\mathrm{(\frac{3}{2}, 0)}\)

7. INFER which choice is NOT an x-intercept

Comparing our results with the answer choices, \(\mathrm{(-\frac{2}{3}, 0)}\) is not among our x-intercepts.

Answer: B. \(\mathrm{(-\frac{2}{3}, 0)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students may misread the question and look for what IS an x-intercept instead of what is NOT an x-intercept. They correctly find the three x-intercepts but then select one of the actual intercepts.

This may lead them to select Choice A \(\mathrm{(-4, 0)}\), Choice C \(\mathrm{(1, 0)}\), or Choice D \(\mathrm{(\frac{3}{2}, 0)}\).

Second Most Common Error:

Inadequate INFER reasoning: Students may not connect that x-intercepts require setting \(\mathrm{f(x) = 0}\), or they may not recognize the need to use zero product property with the factored form. Instead, they might try to expand the polynomial first, making the problem much more difficult.

This leads to confusion and potentially abandoning the systematic approach for guessing.

The Bottom Line:

This problem tests whether students can efficiently use the factored form of a polynomial to find x-intercepts, and whether they can carefully read what the question is actually asking for.