\(\mathrm{g(x) = (x - 2)(x + 1)(x - 5)}\)The function g is given. Which table of values represents \(\mathrm{y =...

1. INFER the key insight about polynomial roots

• Given: \(\mathrm{g(x) = (x - 2)(x + 1)(x - 5)}\) in factored form
• Key insight: When a polynomial is factored, the roots (where \(\mathrm{g(x) = 0}\)) occur when each factor equals zero:
  • \(\mathrm{x - 2 = 0}\) → \(\mathrm{x = 2}\), so \(\mathrm{g(2) = 0}\)
  • \(\mathrm{x + 1 = 0}\) → \(\mathrm{x = -1}\), so \(\mathrm{g(-1) = 0}\)
  • \(\mathrm{x - 5 = 0}\) → \(\mathrm{x = 5}\), so \(\mathrm{g(5) = 0}\)

2. TRANSLATE the transformation requirement

• We need \(\mathrm{y = g(x) + 4}\), not just \(\mathrm{g(x)}\)
• This means: take each \(\mathrm{g(x)}\) value and add 4 to get the corresponding y-value

3. SIMPLIFY the evaluation at each x-value

• At \(\mathrm{x = 2}\): \(\mathrm{g(2) = 0}\), so \(\mathrm{y = 0 + 4 = 4}\)
• At \(\mathrm{x = -1}\): \(\mathrm{g(-1) = 0}\), so \(\mathrm{y = 0 + 4 = 4}\)
• At \(\mathrm{x = 0}\):
  \(\mathrm{g(0) = (0-2)(0+1)(0-5)}\)
  \(\mathrm{= (-2)(1)(-5)}\)
  \(\mathrm{= 10}\)
  so \(\mathrm{y = 10 + 4 = 14}\)

4. INFER which table matches our calculated values

• We need the table showing: \(\mathrm{(2,4), (0,14), (-1,4)}\)
• This matches Choice A exactly

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misread or misunderstand what \(\mathrm{y = g(x) + 4}\) means and think they need to find values for \(\mathrm{g(x)}\) instead of \(\mathrm{g(x) + 4}\).

They correctly identify that \(\mathrm{g(2) = 0}\), \(\mathrm{g(-1) = 0}\), and \(\mathrm{g(0) = 10}\), but forget to add 4 to each value. This leads them to look for a table with values \(\mathrm{(2,0), (0,10), (-1,0)}\).

This may lead them to select Choice B.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when calculating \(\mathrm{g(0) = (0-2)(0+1)(0-5)}\), possibly getting the wrong sign or missing a step in the multiplication.

If they calculate \(\mathrm{g(0)}\) incorrectly as -10 instead of 10, then \(\mathrm{y = -10 + 4 = -6}\), leading them toward an incorrect table choice. This causes confusion and potentially random guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can correctly apply function transformations while managing the details of polynomial evaluation. Success requires both understanding what the transformation means AND executing the arithmetic correctly.