For all real numbers x neq 3, the expression (x^2 - 8x + 21)/(x - 3) can be written in...

1. INFER the problem structure

• Given: \(\mathrm{(x^2 - 8x + 21)/(x - 3)}\) should equal \(\mathrm{x + p + q/(x - 3)}\)
• What this tells us: The form \(\mathrm{x + p + q/(x - 3)}\) suggests we need to divide the polynomial \(\mathrm{x^2 - 8x + 21}\) by \(\mathrm{x - 3}\)
• The result will be: quotient + remainder/(divisor)

2. SIMPLIFY using synthetic division

• Set up synthetic division with \(\mathrm{x = 3}\) (the root of \(\mathrm{x - 3 = 0}\)):

3 | 1 -8 21 |
3 -15 |
1 -5 6

• Bring down the 1
• Multiply: \(\mathrm{1 \times 3 = 3}\), add: \(\mathrm{-8 + 3 = -5}\)
• Multiply: \(\mathrm{-5 \times 3 = -15}\), add: \(\mathrm{21 + (-15) = 6}\)

3. TRANSLATE the division result

• The bottom row gives us: quotient coefficients (1, -5) and remainder (6)
• This means: \(\mathrm{(x^2 - 8x + 21) \div (x - 3) = x - 5}\) with remainder 6
• In fraction form: \(\mathrm{(x^2 - 8x + 21)/(x - 3) = (x - 5) + 6/(x - 3)}\)

4. INFER the values of p and q

• Comparing \(\mathrm{x - 5 + 6/(x - 3)}\) with \(\mathrm{x + p + q/(x - 3)}\):
• \(\mathrm{p = -5}\) and \(\mathrm{q = 6}\)
• Therefore: \(\mathrm{p + q = -5 + 6 = 1}\)

Answer: C (1)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skills: Not recognizing that the target form \(\mathrm{x + p + q/(x - 3)}\) indicates polynomial long division is needed. Students might attempt to factor the numerator or use other algebraic manipulations instead of dividing. This leads to getting stuck and abandoning a systematic approach, causing them to guess.

Second Most Common Error:

Poor SIMPLIFY execution: Making sign errors during synthetic division, especially when handling negative coefficients like -8. Students might get \(\mathrm{p = 5}\) instead of \(\mathrm{p = -5}\), leading to \(\mathrm{p + q = 5 + 6 = 11}\). Since 11 isn't among the choices, this leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can recognize polynomial division patterns and execute the division accurately. The key insight is seeing that the desired form directly tells you what mathematical operation to perform.