x\(\mathrm{p(x)}\)-25-100-31-120The table above gives selected values of a polynomial function p. Based on the values in the table, which of...

1. TRANSLATE the table information

• Given information:
  • Table shows specific x-values and their corresponding \(\mathrm{p(x)}\) values
  • Need to identify which expression must be a factor of polynomial p

2. INFER the connection between zeros and factors

• Look for where \(\mathrm{p(x) = 0}\) in the table—these are the roots/zeros
• From the table: \(\mathrm{p(-1) = 0}\) and \(\mathrm{p(2) = 0}\)
• By the Factor Theorem: if \(\mathrm{p(a) = 0}\), then \(\mathrm{(x - a)}\) is a factor

3. INFER the specific factors from the zeros

• Since \(\mathrm{p(-1) = 0}\), then \(\mathrm{(x - (-1)) = (x + 1)}\) is a factor
• Since \(\mathrm{p(2) = 0}\), then \(\mathrm{(x - 2)}\) is a factor
• Therefore, the product \(\mathrm{(x + 1)(x - 2)}\) is also a factor

4. INFER which answer choice matches

• Check each option against our identified factors:
  • A: \(\mathrm{(x - 3)}\) would need \(\mathrm{x = 3}\) as a root, but no such data
  • B: \(\mathrm{(x + 3)}\) would need \(\mathrm{x = -3}\) as a root, but no such data
  • C: \(\mathrm{(x - 1)(x + 2)}\) would need \(\mathrm{x = 1}\) and \(\mathrm{x = -2}\) as roots, but \(\mathrm{p(1) = -1 ≠ 0}\) and \(\mathrm{p(-2) = 5 ≠ 0}\)
  • D: \(\mathrm{(x + 1)(x - 2)}\) matches exactly with our identified factors ✓

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students correctly identify that \(\mathrm{p(-1) = 0}\) and \(\mathrm{p(2) = 0}\), but get confused about the sign in the factors. They think "if \(\mathrm{x = -1}\) is a root, then \(\mathrm{(x - 1)}\) is a factor" instead of \(\mathrm{(x + 1)}\).

This sign confusion leads them to look for factors like \(\mathrm{(x - 1)}\) and think that \(\mathrm{(x - 1)(x + 2)}\) might work since it has the "right numbers" but wrong signs. This may lead them to select Choice C (\(\mathrm{(x - 1)(x + 2)}\)).

Second Most Common Error:

Missing conceptual knowledge: Students don't remember or understand the Factor Theorem connection between zeros and factors. They might try to substitute the answer choices back into the table values or use some other approach rather than recognizing that \(\mathrm{p(x) = 0}\) directly gives factors.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students can connect the fundamental concept that polynomial zeros correspond to linear factors, with careful attention to the correct signs in those factors.