Which expression is equivalent to \(17(\mathrm{x}^2 - 100\mathrm{y}^2)\)?

1. TRANSLATE the problem information

• Given: \(17(\mathrm{x}^2 - 100\mathrm{y}^2)\)
• Need to find: Equivalent factored expression

2. INFER the factoring approach

• Looking at \(\mathrm{x}^2 - 100\mathrm{y}^2\), I notice this has the form \(\mathrm{a}^2 - \mathrm{b}^2\) (difference of two squares)
• This pattern can be factored using the formula: \(\mathrm{a}^2 - \mathrm{b}^2 = (\mathrm{a} - \mathrm{b})(\mathrm{a} + \mathrm{b})\)
• I need to identify what a and b are in this expression

3. INFER the values of a and b

• First term: \(\mathrm{x}^2\) means \(\mathrm{a} = \mathrm{x}\)
• Second term: \(100\mathrm{y}^2 = (10\mathrm{y})^2\) means \(\mathrm{b} = 10\mathrm{y}\)
• So we have: \(\mathrm{x}^2 - (10\mathrm{y})^2\)

4. SIMPLIFY using the difference of two squares formula

• Apply \(\mathrm{a}^2 - \mathrm{b}^2 = (\mathrm{a} - \mathrm{b})(\mathrm{a} + \mathrm{b})\) with \(\mathrm{a} = \mathrm{x}\) and \(\mathrm{b} = 10\mathrm{y}\)
• \(\mathrm{x}^2 - (10\mathrm{y})^2 = (\mathrm{x} - 10\mathrm{y})(\mathrm{x} + 10\mathrm{y})\)
• Therefore: \(17(\mathrm{x}^2 - 100\mathrm{y}^2) = 17(\mathrm{x} - 10\mathrm{y})(\mathrm{x} + 10\mathrm{y})\)

Answer: D. \(17(\mathrm{x} - 10\mathrm{y})(\mathrm{x} + 10\mathrm{y})\)

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 \(\mathrm{x}^2 - 100\mathrm{y}^2\) and think it needs to be factored like a trinomial \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\), or they might not realize this is a special factoring pattern at all. Without recognizing the \(\mathrm{a}^2 - \mathrm{b}^2\) structure, they get stuck and can't proceed systematically.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor INFER reasoning: Incorrectly identifying the square roots

Students recognize the difference of two squares pattern but make errors finding a and b. They might think \(\sqrt{100\mathrm{y}^2} = 2\mathrm{y}\) (confusing \(\sqrt{100} = 2\) instead of 10), leading them to try factoring as \((\mathrm{x} - 2\mathrm{y})(\mathrm{x} + 2\mathrm{y})\), which doesn't match any choice exactly. Or they might get confused about the second term and end up with something like \((\mathrm{x} - 2\mathrm{y})(\mathrm{x} - 50\mathrm{y})\) or \((\mathrm{x} - 2\mathrm{y})(\mathrm{x} + 50\mathrm{y})\).

This may lead them to select Choice A (\(17(\mathrm{x} - 2\mathrm{y})(\mathrm{x} - 50\mathrm{y})\)) or Choice B (\(17(\mathrm{x} - 2\mathrm{y})(\mathrm{x} + 50\mathrm{y})\)).

The Bottom Line:

Success on this problem depends on pattern recognition - specifically identifying when you have a difference of two squares versus other polynomial forms, then correctly finding the square roots of both terms.