Which expression is equivalent to 256w^2 - 676?

1. INFER the factoring approach

• Look at the expression: \(256\mathrm{w}^2 - 676\)
• This is a subtraction of two terms, so check if it's a difference of two squares pattern
• Recall: \(\mathrm{a}^2 - \mathrm{b}^2 = (\mathrm{a} - \mathrm{b})(\mathrm{a} + \mathrm{b})\)

2. SIMPLIFY to identify the perfect squares

• Check if \(256\mathrm{w}^2\) is a perfect square:
  • \(\sqrt{256} = 16\), so \(256\mathrm{w}^2 = (16\mathrm{w})^2\)
• Check if 676 is a perfect square:
  • \(\sqrt{676} = 26\), so \(676 = 26^2\)
• Now we have: \((16\mathrm{w})^2 - 26^2\)

3. SIMPLIFY using the difference of squares formula

• Apply \(\mathrm{a}^2 - \mathrm{b}^2 = (\mathrm{a} - \mathrm{b})(\mathrm{a} + \mathrm{b})\) where \(\mathrm{a} = 16\mathrm{w}\) and \(\mathrm{b} = 26\)
• \((16\mathrm{w})^2 - 26^2 = (16\mathrm{w} - 26)(16\mathrm{w} + 26)\)

Answer: D. \((16\mathrm{w} - 26)(16\mathrm{w} + 26)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the difference of two squares pattern

Students might see \(256\mathrm{w}^2 - 676\) and attempt to factor by grouping or look for common factors instead of identifying this as \(\mathrm{a}^2 - \mathrm{b}^2\). Without this key insight, they get stuck and resort to guessing or trying to work backwards from answer choices.

This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Incorrectly finding square roots

Students might correctly identify the pattern but make arithmetic errors: thinking \(\sqrt{256} = 8\) instead of 16, or \(\sqrt{676} = 13\) instead of 26. With \(\sqrt{256} = 8\), they would think \(256\mathrm{w}^2 = (8\mathrm{w})^2\), leading them to factor as \((8\mathrm{w} - 13)(8\mathrm{w} + 13)\).

This may lead them to select Choice B (\((8\mathrm{w} - 13)(8\mathrm{w} + 13)\)).

The Bottom Line:

This problem tests pattern recognition first, then computational accuracy. Students who don't immediately see the difference of squares structure often struggle, while those who see the pattern but make arithmetic errors in finding square roots will select plausible but incorrect factorizations.