Which of the following expressions is (or are) a factor of x^3 + 4x^2 + x + 4? x -...

1. INFER the testing strategy

• Since we need to check if linear expressions are factors of a polynomial, the Factor Theorem is our tool
• Factor Theorem: \(\mathrm{(x - k)}\) is a factor of \(\mathrm{f(x)}\) if and only if \(\mathrm{f(k) = 0}\)
• This means we substitute specific values into the polynomial and check if we get zero

2. SIMPLIFY the test for Expression I: x - 1

• For \(\mathrm{x - 1}\) to be a factor, we need \(\mathrm{f(1) = 0}\)
• \(\mathrm{f(1) = (1)^3 + 4(1)^2 + (1) + 4}\)
• \(\mathrm{f(1) = 1 + 4 + 1 + 4 = 10}\)
• Since \(\mathrm{f(1) = 10 \neq 0}\), expression I is NOT a factor

3. SIMPLIFY the test for Expression II: x + 4

• Note that \(\mathrm{x + 4 = x - (-4)}\), so we need \(\mathrm{f(-4) = 0}\)
• \(\mathrm{f(-4) = (-4)^3 + 4(-4)^2 + (-4) + 4}\)
• \(\mathrm{f(-4) = -64 + 4(16) - 4 + 4}\)
• \(\mathrm{f(-4) = -64 + 64 - 4 + 4 = 0}\)
• Since \(\mathrm{f(-4) = 0}\), expression II IS a factor

4. INFER the final answer

• Only expression II \(\mathrm{(x + 4)}\) is a factor
• This corresponds to answer choice B

Answer: B) II only

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when evaluating the polynomial, especially with negative numbers in \(\mathrm{f(-4)}\).

The calculation \(\mathrm{f(-4) = (-4)^3 + 4(-4)^2 + (-4) + 4}\) involves several sign changes and can easily lead to errors like:

• Forgetting that \(\mathrm{(-4)^3 = -64}\) (not +64)
• Miscalculating \(\mathrm{4(-4)^2}\) as -64 instead of +64
• Sign errors in the final addition: \(\mathrm{-64 + 64 - 4 + 4}\)

These arithmetic mistakes lead to concluding that \(\mathrm{f(-4) \neq 0}\), causing them to think \(\mathrm{x + 4}\) is not a factor. This may lead them to select Choice D (Neither I nor II).

Second Most Common Error:

Poor INFER reasoning about the Factor Theorem application: Students might try to factor the polynomial algebraically instead of using substitution.

They attempt methods like grouping: \(\mathrm{x^3 + 4x^2 + x + 4 = x^2(x + 4) + 1(x + 4) = (x^2 + 1)(x + 4)}\), which is correct but more complex. However, they might make errors in the factoring process or not recognize that this confirms \(\mathrm{x + 4}\) as a factor while showing \(\mathrm{x - 1}\) is not. This leads to confusion and guessing.

The Bottom Line:

The Factor Theorem provides a direct computational path, but success depends on careful arithmetic with negative numbers and systematic testing of each option.