Which of the following is equivalent to \((1 - \mathrm{p})(1 + \mathrm{p} + \mathrm{p}^2 + \mathrm{p}^3 + \mathrm{p}^4 + \mathrm{p}^5...

1. TRANSLATE the problem information

• Given expression: \((1 - \mathrm{p})(1 + \mathrm{p} + \mathrm{p}^2 + \mathrm{p}^3 + \mathrm{p}^4 + \mathrm{p}^5 + \mathrm{p}^6)\)
• Need to find: Which answer choice this expression equals

2. INFER the approach

• The binomial \((1 - \mathrm{p})\) needs to be distributed across every term in the polynomial
• This will create many terms that we'll need to combine systematically
• Look for patterns in the resulting expression

3. SIMPLIFY through distribution

Distribute \((1 - \mathrm{p})\) to get two separate parts:

• First part: \(1(1 + \mathrm{p} + \mathrm{p}^2 + \mathrm{p}^3 + \mathrm{p}^4 + \mathrm{p}^5 + \mathrm{p}^6) = 1 + \mathrm{p} + \mathrm{p}^2 + \mathrm{p}^3 + \mathrm{p}^4 + \mathrm{p}^5 + \mathrm{p}^6\)
• Second part: \(-\mathrm{p}(1 + \mathrm{p} + \mathrm{p}^2 + \mathrm{p}^3 + \mathrm{p}^4 + \mathrm{p}^5 + \mathrm{p}^6) = -\mathrm{p} - \mathrm{p}^2 - \mathrm{p}^3 - \mathrm{p}^4 - \mathrm{p}^5 - \mathrm{p}^6 - \mathrm{p}^7\)

4. SIMPLIFY by combining like terms

Write out all terms:

\(1 + \mathrm{p} + \mathrm{p}^2 + \mathrm{p}^3 + \mathrm{p}^4 + \mathrm{p}^5 + \mathrm{p}^6 - \mathrm{p} - \mathrm{p}^2 - \mathrm{p}^3 - \mathrm{p}^4 - \mathrm{p}^5 - \mathrm{p}^6 - \mathrm{p}^7\)

5. INFER the telescoping pattern

Notice that most terms cancel:

• \(+\mathrm{p}\) and \(-\mathrm{p}\) cancel
• \(+\mathrm{p}^2\) and \(-\mathrm{p}^2\) cancel
• \(+\mathrm{p}^3\) and \(-\mathrm{p}^3\) cancel
• \(+\mathrm{p}^4\) and \(-\mathrm{p}^4\) cancel
• \(+\mathrm{p}^5\) and \(-\mathrm{p}^5\) cancel
• \(+\mathrm{p}^6\) and \(-\mathrm{p}^6\) cancel

Only \(1\) and \(-\mathrm{p}^7\) remain: \(1 - \mathrm{p}^7\)

Answer: B. \(1 - \mathrm{p}^7\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor SIMPLIFY execution: Students make distribution errors, particularly with the negative sign from \((1 - \mathrm{p})\). They might forget to distribute the negative to all terms, writing something like:

\(1 + \mathrm{p} + \mathrm{p}^2 + \mathrm{p}^3 + \mathrm{p}^4 + \mathrm{p}^5 + \mathrm{p}^6 - \mathrm{p} + \mathrm{p}^2 + \mathrm{p}^3 + \mathrm{p}^4 + \mathrm{p}^5 + \mathrm{p}^6 + \mathrm{p}^7\)

This creates wrong signs that don't cancel properly, leading to expressions like \(1 + 2\mathrm{p} + 2\mathrm{p}^2 + ... + \mathrm{p}^7\) or other incorrect forms. This may lead them to select Choice A (\(1 - \mathrm{p}^8\)) or get confused and guess.

Second Most Common Error:

Weak INFER skills: Students don't recognize the telescoping pattern and try to combine terms incorrectly. They might group terms by degree instead of recognizing the cancellation pattern, or stop too early in the simplification process. This leads to confusion and guessing rather than systematic completion.

The Bottom Line:

This problem tests whether students can execute systematic polynomial multiplication while recognizing elegant cancellation patterns. The key insight is seeing that distribution creates perfectly matched positive and negative terms that cancel, leaving only the 'end' terms.