Which of the following is equivalent to the expression \(4(\mathrm{y}^2 + 2\mathrm{z}) - (\mathrm{y}^2 + 2\mathrm{z})\)?

1. TRANSLATE the problem information

• Given expression: \(\mathrm{4(y^2 + 2z) - (y^2 + 2z)}\)
• Goal: Find an equivalent simplified form

2. INFER the best approach

• Notice both terms contain the same expression \(\mathrm{(y^2 + 2z)}\)
• Two viable approaches: factor out the common term, or distribute first
• Either method will work - let's use the factoring method as it's more efficient

3. SIMPLIFY by factoring the common term

• Rewrite as: \(\mathrm{4(y^2 + 2z) - 1(y^2 + 2z)}\)
• Factor out \(\mathrm{(y^2 + 2z)}\): \(\mathrm{(4 - 1)(y^2 + 2z) = 3(y^2 + 2z)}\)

4. SIMPLIFY by distributing

• Apply distributive property: \(\mathrm{3(y^2 + 2z) = 3y^2 + 3(2z) = 3y^2 + 6z}\)

Answer: B (\(\mathrm{3y^2 + 6z}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly handle the negative sign when distributing.

Instead of treating \(\mathrm{-(y^2 + 2z)}\) as \(\mathrm{-1(y^2 + 2z) = -y^2 - 2z}\), they might write it as \(\mathrm{-y^2 + 2z}\). This leads to:

\(\mathrm{4y^2 + 8z - y^2 + 2z = 3y^2 + 10z}\)

This may lead them to select Choice C (\(\mathrm{3y^2 + 10z}\))

Second Most Common Error:

Poor INFER reasoning: Students don't recognize the common factor pattern and get lost in the algebra.

They might start distributing but make arithmetic errors in combining coefficients, or they might not see that both terms have the same binomial factor. This confusion can cause calculation mistakes throughout the problem.

This leads to confusion and potentially selecting any of the incorrect answer choices.

The Bottom Line:

This problem tests whether students can recognize common algebraic patterns and execute multi-step simplification accurately. The key insight is seeing that \(\mathrm{(y^2 + 2z)}\) appears in both terms, making factoring the most elegant approach.