If \(\mathrm{f(a) = 3a^{2}(a - 4)}\), what is the coefficient of a^(2) in the expanded form of \(\mathrm{f(a)}\)?

1. TRANSLATE the problem information

• Given: \(\mathrm{f(a) = 3a^2(a - 4)}\)
• Find: coefficient of \(\mathrm{a^2}\) in the expanded form

2. INFER the solution approach

• To find the coefficient of \(\mathrm{a^2}\), we need to expand the expression first
• The expression \(\mathrm{3a^2(a - 4)}\) is in factored form and needs to be distributed

3. SIMPLIFY using the distributive property

• Apply the distributive property: \(\mathrm{3a^2(a - 4) = 3a^2 \cdot a + 3a^2 \cdot (-4)}\)
• Multiply the first term: \(\mathrm{3a^2 \cdot a = 3a^3}\) (using \(\mathrm{a^m \cdot a^n = a^{m+n}}\))
• Multiply the second term: \(\mathrm{3a^2 \cdot (-4) = -12a^2}\)
• Combined result: \(\mathrm{f(a) = 3a^3 - 12a^2}\)

4. INFER the final answer

• In the expanded form \(\mathrm{f(a) = 3a^3 - 12a^2}\), the coefficient of \(\mathrm{a^2}\) is \(\mathrm{-12}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students incorrectly apply the distributive property or make errors with exponent rules.

They might write: \(\mathrm{3a^2(a - 4) = 3a^2 + a - 4}\) (forgetting to distribute to both terms) or \(\mathrm{3a^2(a - 4) = 3a^2 - 12a^2}\) (forgetting to multiply \(\mathrm{3a^2}\) by \(\mathrm{a}\)). These algebraic errors lead to wrong expanded forms.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Conceptual confusion about coefficients: Students identify the wrong coefficient from their expansion.

Even if they correctly expand to get \(\mathrm{3a^3 - 12a^2}\), they might focus on the \(\mathrm{3}\) (coefficient of \(\mathrm{a^3}\)) or just the \(\mathrm{4}\) from the original expression. They might think "the coefficient" refers to any numerical part they see.

This may lead them to select Choice B (3) or Choice A (-4).

The Bottom Line:

This problem requires careful algebraic manipulation combined with precise terminology understanding. Students must both execute the distributive property correctly AND identify the specific coefficient requested.