Which of the following is equivalent to \((2\mathrm{x} + \frac{\mathrm{y}}{3})^2\)?

1. TRANSLATE the problem information

• Given expression: \((2x + \frac{y}{3})^2\)
• This is a perfect square binomial that requires expansion

2. INFER the approach

• Use the binomial expansion formula: \((a + b)^2 = a^2 + 2ab + b^2\)
• Identify: \(a = 2x\) and \(b = \frac{y}{3}\)
• We'll need to calculate each term separately then combine

3. SIMPLIFY each term of the expansion

First term: \(a^2 = (2x)^2\)

• \((2x)^2 = 2^2 \times x^2 = 4x^2\)

Middle term: \(2ab = 2(2x)(\frac{y}{3})\)

• \(2(2x)(\frac{y}{3}) = 2 \times 2x \times \frac{y}{3} = \frac{4xy}{3}\)

Last term: \(b^2 = (\frac{y}{3})^2\)

• \((\frac{y}{3})^2 = \frac{y^2}{3^2} = \frac{y^2}{9}\)

4. SIMPLIFY by combining all terms

• Final expression: \(4x^2 + \frac{4xy}{3} + \frac{y^2}{9}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students forget the coefficient "2" in the middle term \(2ab\)

Many students remember the pattern \(a^2 + ? + b^2\) but forget that the middle term needs the factor of 2. They calculate the middle term as just \((2x)(\frac{y}{3}) = \frac{2xy}{3}\) instead of \(2(2x)(\frac{y}{3}) = \frac{4xy}{3}\).

This leads them to select Choice B \((4x^2 + \frac{2xy}{3} + \frac{y^2}{9})\)

Second Most Common Error:

Missing conceptual knowledge: Not knowing the binomial expansion formula

Some students attempt to multiply \((2x + \frac{y}{3})(2x + \frac{y}{3})\) using FOIL but make errors in the process, or they simply don't know any systematic approach for expanding squared binomials.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

The key challenge is accurately applying the binomial expansion formula while carefully handling the arithmetic, especially ensuring the middle term has the correct coefficient of 2.