Let \(\mathrm{h(x) = (x + 1)(x - 2)(x + 4)(2x - 3)}\). In the xy-plane, which of the following is...

1. TRANSLATE the problem information

• Given: \(\mathrm{h(x) = (x + 1)(x - 2)(x + 4)(2x - 3)}\)
• Find: Which answer choice is NOT an x-intercept
• What this tells us: X-intercepts occur where the graph crosses the x-axis, meaning where \(\mathrm{y = h(x) = 0}\)

2. INFER the solution strategy

• Since \(\mathrm{h(x)}\) is already in factored form, we can use the Zero Product Property
• Key insight: A product equals zero if and only if at least one factor equals zero
• Strategy: Set each factor equal to zero and solve

3. SIMPLIFY by solving each equation

Set each factor equal to zero:

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

4. APPLY CONSTRAINTS to identify the answer

• The x-intercepts are at \(\mathrm{x = -1}\), \(\mathrm{x = 2}\), \(\mathrm{x = -4}\), and \(\mathrm{x = \frac{3}{2}}\)
• Check each answer choice:
  • (A) (-4, 0): \(\mathrm{x = -4}\) ✓ IS an intercept
  • (B) (-1, 0): \(\mathrm{x = -1}\) ✓ IS an intercept
  • (C) (1, 0): \(\mathrm{x = 1}\) ✗ NOT an intercept
  • (D) (3/2, 0): \(\mathrm{x = \frac{3}{2}}\) ✓ IS an intercept

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not clearly connect 'x-intercept' with the condition \(\mathrm{y = 0}\), instead trying to substitute the given points directly into \(\mathrm{h(x)}\) to check if they work.

For example, they might substitute \(\mathrm{x = 1}\) into:

\(\mathrm{h(x) = (1 + 1)(1 - 2)(1 + 4)(2(1) - 3)}\)
\(\mathrm{= (2)(-1)(5)(-1)}\)
\(\mathrm{= 10 \neq 0}\)

then get confused about what this means. This leads to uncertainty and potentially guessing between answers.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the zero-factor equations but make algebraic errors, particularly with \(\mathrm{2x - 3 = 0}\). They might incorrectly solve this as \(\mathrm{x = \frac{3}{2} = 1.5}\), then think \(\mathrm{x = 1}\) is 'close enough,' or make sign errors in other equations.

This may lead them to select Choice C (1, 0) thinking it's actually correct, or become confused about which intercepts are real.

The Bottom Line:

This problem tests whether students can connect the graphical concept of x-intercepts with the algebraic condition of setting a function equal to zero, then execute the Zero Product Property correctly.