Which of the following expressions is equivalent to (x^2 - 16)/(x - 4) for x neq 4?
GMAT Advanced Math : (Adv_Math) Questions
Which of the following expressions is equivalent to \(\frac{\mathrm{x}^2 - 16}{\mathrm{x} - 4}\) for \(\mathrm{x} \neq 4\)?
1. INFER the factoring opportunity
- Looking at the numerator \(\mathrm{x^2 - 16}\), recognize this matches the difference of squares pattern
- Pattern: \(\mathrm{a^2 - b^2 = (a + b)(a - b)}\)
- Here: \(\mathrm{x^2 - 16 = x^2 - 4^2 = (x + 4)(x - 4)}\)
2. SIMPLIFY by substituting the factored form
- Replace the numerator with its factored form:
\(\frac{\mathrm{x^2 - 16}}{\mathrm{x - 4}} = \frac{\mathrm{(x + 4)(x - 4)}}{\mathrm{x - 4}}\)
3. SIMPLIFY by canceling common factors
- Since \(\mathrm{x \neq 4}\), we know \(\mathrm{(x - 4) \neq 0}\), so we can cancel:
\(\frac{\mathrm{(x + 4)(x - 4)}}{\mathrm{x - 4}} = \mathrm{x + 4}\) - The restriction \(\mathrm{x \neq 4}\) ensures this cancellation is mathematically valid
Answer: C (\(\mathrm{x + 4}\))
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Not recognizing \(\mathrm{x^2 - 16}\) as a difference of squares
Students see \(\mathrm{x^2 - 16}\) and don't immediately connect it to the factoring pattern \(\mathrm{a^2 - b^2 = (a + b)(a - b)}\). Without this insight, they can't factor the numerator and become stuck trying to simplify the expression directly. This leads to confusion and guessing among the answer choices.
Second Most Common Error:
Incomplete SIMPLIFY execution: Attempting polynomial long division instead of factoring
Some students recognize they need to simplify but choose polynomial long division rather than factoring. While this approach can work, it's more complex and error-prone. Students might make arithmetic errors during division or not complete the process correctly, potentially leading them to select Choice E (\(\mathrm{x^2 - 4}\)) if they stop partway through.
The Bottom Line:
The key insight is recognizing the difference of squares pattern in the numerator. Once you see \(\mathrm{x^2 - 16 = (x + 4)(x - 4)}\), the solution becomes straightforward cancellation. Students who miss this pattern struggle to make progress on the problem.