Which of the following expressions is equivalent to (x^2 - 2x - 5)/(x - 3)?

1. INFER the solution approach

• We have a rational expression \(\frac{\mathrm{x}^2 - 2\mathrm{x} - 5}{\mathrm{x} - 3}\) that needs to be rewritten
• Since the numerator is a quadratic and denominator is linear, we can use polynomial long division
• The goal is to express this as: \(\mathrm{quotient} + \frac{\mathrm{remainder}}{\mathrm{x} - 3}\)

2. SIMPLIFY using polynomial long division

First division step:

• Divide the leading term: \(\mathrm{x}^2 \div \mathrm{x} = \mathrm{x}\)
• Multiply back: \(\mathrm{x}(\mathrm{x} - 3) = \mathrm{x}^2 - 3\mathrm{x}\)
• Subtract: \((\mathrm{x}^2 - 2\mathrm{x} - 5) - (\mathrm{x}^2 - 3\mathrm{x}) = \mathrm{x} - 5\)

Second division step:

• Divide the leading term: \(\mathrm{x} \div \mathrm{x} = 1\)
• Multiply back: \(1(\mathrm{x} - 3) = \mathrm{x} - 3\)
• Subtract: \((\mathrm{x} - 5) - (\mathrm{x} - 3) = -2\)

3. SIMPLIFY to write the final form

• We have: \(\mathrm{quotient} = \mathrm{x} + 1\), \(\mathrm{remainder} = -2\)
• Therefore: \(\frac{\mathrm{x}^2 - 2\mathrm{x} - 5}{\mathrm{x} - 3} = \mathrm{x} + 1 + \frac{-2}{\mathrm{x} - 3}\)
• Which simplifies to: \(\mathrm{x} + 1 - \frac{2}{\mathrm{x} - 3}\)

Answer: D. \(\mathrm{x} + 1 - \frac{2}{\mathrm{x} - 3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during polynomial long division, particularly when subtracting polynomials or handling negative terms.

For example, when subtracting \((\mathrm{x}^2 - 3\mathrm{x})\) from \((\mathrm{x}^2 - 2\mathrm{x} - 5)\), they might get \(\mathrm{x} + 5\) instead of \(\mathrm{x} - 5\), or make sign errors with the remainder. These calculation mistakes lead to incorrect coefficients in the final answer.

This may lead them to select Choice A \((\mathrm{x} - 5 - \frac{20}{\mathrm{x}-3})\) or Choice B \((\mathrm{x} - 5 - \frac{10}{\mathrm{x}-3})\).

Second Most Common Error:

Inadequate INFER reasoning: Students don't recognize that polynomial long division is the appropriate strategy, instead attempting to factor or use other algebraic manipulations that don't work efficiently.

Without a clear strategy, they may attempt to expand answer choices backwards or guess based on partial pattern recognition, leading to confusion and random answer selection.

The Bottom Line:

This problem tests whether students can execute polynomial long division accurately while maintaining careful attention to arithmetic details. The multiple steps and potential for sign errors make systematic calculation essential for success.