\(\mathrm{f(x) = x^3 + 3x^2 - 6x - 1}\). For the function f defined above, what is the value of...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = x^3 + 3x^2 - 6x - 1}\)
  • Need to find: \(\mathrm{f(-1)}\)
• What this tells us: We need to substitute \(\mathrm{x = -1}\) everywhere x appears in the function

2. SIMPLIFY by substituting and evaluating

• Replace every x with (-1):
  \(\mathrm{f(-1) = (-1)^3 + 3(-1)^2 - 6(-1) - 1}\)

• Evaluate each term carefully:
  • \(\mathrm{(-1)^3 = -1}\) (negative base cubed stays negative)
  • \(\mathrm{3(-1)^2 = 3(1) = 3}\) (negative base squared becomes positive)
  • \(\mathrm{-6(-1) = 6}\) (negative times negative is positive)
  • \(\mathrm{-1 = -1}\) (constant term)

3. SIMPLIFY by combining terms

• \(\mathrm{f(-1) = -1 + 3 + 6 - 1}\)
• \(\mathrm{f(-1) = 7}\)

Answer: C. 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students confuse the signs when evaluating powers of negative numbers, especially mixing up \(\mathrm{(-1)^3 = -1}\) and \(\mathrm{(-1)^2 = 1}\).

A common mistake is treating \(\mathrm{(-1)^3}\) as positive 1, leading to \(\mathrm{f(-1) = 1 + 3 + 6 + 1 = 11}\). This may lead them to select Choice D (11).

Second Most Common Error:

Poor SIMPLIFY reasoning: Students make multiple sign errors throughout the calculation, particularly with \(\mathrm{-6(-1)}\) or the final combination step.

One pattern is calculating \(\mathrm{f(-1) = -1 - 3 - 6 - 1 = -11}\) by incorrectly handling the negative coefficients. This may lead them to select Choice A (-11).

The Bottom Line:

This problem tests careful arithmetic with negative numbers more than advanced function concepts. The key is methodically evaluating each term without rushing through the signs.