When polynomial \(\mathrm{f(x)}\) is divided by \(\mathrm{(x - 3)}\), the remainder is 0. When polynomial \(\mathrm{f(x)}\) is divided by \(\mathrm{(x...

1. TRANSLATE the division information into function values

• Given information:
  • \(\mathrm{f(x)}\) ÷ \(\mathrm{(x - 3)}\) has remainder 0
  • \(\mathrm{f(x)}\) ÷ \(\mathrm{(x + 2)}\) has remainder 0
  • \(\mathrm{f(x)}\) ÷ \(\mathrm{(x - 1)}\) has remainder 4

• Using the Remainder Theorem: When \(\mathrm{f(x)}\) is divided by \(\mathrm{(x - a)}\), the remainder equals \(\mathrm{f(a)}\)

2. INFER what these remainders tell us about factors

• \(\mathrm{f(3) = 0}\) means \(\mathrm{(x - 3)}\) is a factor of \(\mathrm{f(x)}\)
• \(\mathrm{f(-2) = 0}\) means \(\mathrm{(x + 2)}\) is a factor of \(\mathrm{f(x)}\)
• \(\mathrm{f(1) = 4 ≠ 0}\) means \(\mathrm{(x - 1)}\) is NOT a factor of \(\mathrm{f(x)}\)

3. INFER which answer choice represents a guaranteed factor

• Choice A: \(\mathrm{(x - 4)}\) - We have no information about \(\mathrm{f(4)}\), so this isn't guaranteed
• Choice B: \(\mathrm{(x - 1)}\) - We know \(\mathrm{f(1) ≠ 0}\), so this is definitely not a factor
• Choice C: \(\mathrm{(x - 3)(x + 2)}\) - Both individual factors are confirmed, so their product is guaranteed to be a factor
• Choice D: \(\mathrm{(x - 3)(x - 1)}\) - Contains \(\mathrm{(x - 1)}\) which we know is not a factor

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not understanding the connection between remainder and factors

Students might think "remainder 4 when divided by \(\mathrm{(x - 1)}\)" somehow makes \(\mathrm{(x - 1)}\) a factor, confusing the Factor Theorem requirement that the remainder must be zero. They miss that \(\mathrm{f(1) = 4 ≠ 0}\) means \(\mathrm{(x - 1)}\) is definitely NOT a factor.

This may lead them to select Choice B (\(\mathrm{(x - 1)}\)) or Choice D (\(\mathrm{(x - 3)(x - 1)}\)).

Second Most Common Error:

Missing conceptual knowledge: Not remembering the Remainder Theorem

Students might recognize that remainders of 0 suggest factors, but without the Remainder Theorem, they can't systematically translate "divided by \(\mathrm{(x + 2)}\) with remainder 0" into "\(\mathrm{f(-2) = 0}\)." This leads to guessing based on partial understanding.

This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether you can move fluidly between division language and function evaluation language using the Remainder Theorem, then apply logical reasoning about what combinations of factors are guaranteed to exist.