9x^2 - 121y^2 Which of the following is a factor of the polynomial above?...

1. INFER the pattern type

• Look at the structure: \(9\mathrm{x}^2 - 121\mathrm{y}^2\)
• This is two terms separated by subtraction
• Both terms are perfect squares
• This matches the difference of squares pattern: \(\mathrm{a}^2 - \mathrm{b}^2\)

2. INFER the perfect square components

• First term: \(9\mathrm{x}^2 = (3\mathrm{x})^2\) → so \(\mathrm{a} = 3\mathrm{x}\)
• Second term: \(121\mathrm{y}^2 = (11\mathrm{y})^2\) → so \(\mathrm{b} = 11\mathrm{y}\)
• Key insight: We need the square root of each coefficient (\(\sqrt{9} = 3\), \(\sqrt{121} = 11\))

3. SIMPLIFY using the difference of squares formula

• Apply: \(\mathrm{a}^2 - \mathrm{b}^2 = (\mathrm{a} - \mathrm{b})(\mathrm{a} + \mathrm{b})\)
• Substitute our values: \((3\mathrm{x})^2 - (11\mathrm{y})^2 = (3\mathrm{x} - 11\mathrm{y})(3\mathrm{x} + 11\mathrm{y})\)
• The two factors are \((3\mathrm{x} - 11\mathrm{y})\) and \((3\mathrm{x} + 11\mathrm{y})\)

4. INFER which factor appears in the choices

• Looking at the answer choices, \((3\mathrm{x} - 11\mathrm{y})\) appears as choice (A)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the difference of squares pattern and instead try to factor by grouping or other methods that don't apply here.

They might see \(9\mathrm{x}^2 - 121\mathrm{y}^2\) and think they need to find common factors or try other factoring techniques. Without recognizing this specific pattern, they get stuck and end up guessing among the choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students recognize the pattern but make errors identifying the square root values, particularly thinking \(\sqrt{121} = 121\) instead of \(\sqrt{121} = 11\).

This leads them to incorrectly identify \(\mathrm{b} = 121\mathrm{y}\) instead of \(\mathrm{b} = 11\mathrm{y}\), causing them to look for factors like \((3\mathrm{x} + 121\mathrm{y})\). This may lead them to select Choice B \((3\mathrm{x} + 121\mathrm{y})\).

The Bottom Line:

Success on this problem hinges on pattern recognition - specifically seeing that this is a difference of squares and knowing that both terms must be perfect squares for the formula to apply.