A polynomial function is defined by the equation \(\mathrm{f(x) = \frac{1}{2} \cdot (x - 2)(x + 7)(x - 11)}\). The...

1. TRANSLATE the problem information

• Given: \(\mathrm{f(x) = \frac{1}{2} \cdot (x - 2)(x + 7)(x - 11)}\)
• Find: Sum of x-coordinates of x-intercepts
• What this means: Find where the graph crosses the x-axis, then add those x-values

2. TRANSLATE x-intercepts to mathematical condition

• X-intercepts occur where \(\mathrm{f(x) = 0}\)
• Set up the equation: \(\mathrm{0 = \frac{1}{2} \cdot (x - 2)(x + 7)(x - 11)}\)

3. INFER the most efficient approach

• The function is already in factored form - this is perfect for finding zeros!
• Since \(\mathrm{\frac{1}{2} \neq 0}\), we can use the zero product property
• The product equals zero when any individual factor equals zero

4. SIMPLIFY by solving each factor

• Set each factor equal to zero:
  • \(\mathrm{x - 2 = 0 \rightarrow x = 2}\)
  • \(\mathrm{x + 7 = 0 \rightarrow x = -7}\)
  • \(\mathrm{x - 11 = 0 \rightarrow x = 11}\)

5. SIMPLIFY to find the final sum

• Sum = \(\mathrm{2 + (-7) + 11 = 6}\)

Answer: C) 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misunderstand what "x-intercepts" means and try to substitute specific x-values instead of setting \(\mathrm{f(x) = 0}\).

They might substitute \(\mathrm{x = 0}\) or try to find the y-intercept instead, leading to confusion about what the problem is actually asking. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly find the x-intercepts but make arithmetic errors when computing the sum.

For example, they might calculate \(\mathrm{2 + (-7) + 11}\)

\(\mathrm{= 2 - 7 + 11}\)

\(\mathrm{= -5 + 11}\)

\(\mathrm{= 16}\)

instead of 6, or forget the negative sign on -7. This may lead them to select Choice A) -16 or other incorrect values.

The Bottom Line:

The key insight is recognizing that the factored form makes this problem straightforward - you don't need to expand or do complex algebra. The challenge lies in correctly translating "x-intercepts" to the condition \(\mathrm{f(x) = 0}\) and then executing the arithmetic carefully.