The values of a function g, defined by \(\mathrm{g(x) = f(x - 1)}\) where f is a polynomial, are given...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{g(x) = f(x-1)}\) where \(\mathrm{f}\) is a polynomial
  • \(\mathrm{g(-1) = 0}\) and \(\mathrm{g(3) = 0}\) (among other values)
• What this tells us: We need to find factors of \(\mathrm{f}\), but we're given information about \(\mathrm{g}\)

2. INFER the key relationship

• Since \(\mathrm{g(x) = f(x-1)}\), when \(\mathrm{g}\) equals zero, \(\mathrm{f}\) must also equal zero
• If \(\mathrm{g(a) = 0}\), then \(\mathrm{f(a-1) = 0}\) by direct substitution
• This is our bridge from information about \(\mathrm{g}\) to information about \(\mathrm{f}\)

3. APPLY the relationship to find f's zeros

• \(\mathrm{g(-1) = 0}\) means \(\mathrm{f(-1-1) = f(-2) = 0}\)
• \(\mathrm{g(3) = 0}\) means \(\mathrm{f(3-1) = f(2) = 0}\)
• So \(\mathrm{f}\) has zeros at \(\mathrm{x = -2}\) and \(\mathrm{x = 2}\)

4. INFER the factors from the zeros

• Since \(\mathrm{f}\) is a polynomial with zero at \(\mathrm{x = -2}\), then \(\mathrm{(x-(-2)) = (x+2)}\) is a factor
• Since \(\mathrm{f}\) is a polynomial with zero at \(\mathrm{x = 2}\), then \(\mathrm{(x-2)}\) is a factor
• Both factors must be present in \(\mathrm{f}\)

5. APPLY CONSTRAINTS to select the answer

• Looking at answer choices, only (D) contains both required factors: \(\mathrm{(x-2)(x+2)}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not making the connection between \(\mathrm{g(x) = f(x-1)}\) and the zero relationship.

Students see that \(\mathrm{g(-1) = 0}\) and \(\mathrm{g(3) = 0}\), but don't realize this means \(\mathrm{f(-2) = 0}\) and \(\mathrm{f(2) = 0}\). They might try to work directly with the x-values where \(\mathrm{g}\) is zero, thinking \(\mathrm{f}\) has zeros at \(\mathrm{x = -1}\) and \(\mathrm{x = 3}\). This leads them to look for factors like \(\mathrm{(x+1)}\) and \(\mathrm{(x-3)}\), which aren't among the choices, causing confusion and guessing.

Second Most Common Error:

Incomplete INFER reasoning: Finding only one factor instead of recognizing both are needed.

Students might correctly identify that \(\mathrm{f(-2) = 0}\) so \(\mathrm{(x+2)}\) is a factor, but miss that \(\mathrm{f(2) = 0}\) so \(\mathrm{(x-2)}\) is also a factor. They see \(\mathrm{(x+2)}\) appears in choice (E), so they select that without checking if all the zeros are accounted for. This may lead them to select Choice E (\(\mathrm{(x-1)(x+2)}\)).

The Bottom Line:

The key insight is translating between function compositions - when the composed function \(\mathrm{g}\) has a zero, you need to "work backwards" through the composition to find where the inner function \(\mathrm{f}\) has its zero.