Let the function f be defined by \(\mathrm{f(x) = 3x^2 - 5x + 4}\), and let the function g be...

1. TRANSLATE the problem information

• Given functions:
  • \(\mathrm{f(x) = 3x^2 - 5x + 4}\)
  • \(\mathrm{g(x) = 6x^2 + 20x - 13}\)
• Need to find: A factor of \(\mathrm{f(x) + g(x)}\)

2. SIMPLIFY by adding the polynomials

• Add the two functions:
  \(\mathrm{f(x) + g(x) = (3x^2 - 5x + 4) + (6x^2 + 20x - 13)}\)

• Combine like terms carefully:
  • \(\mathrm{x^2}\) terms: \(\mathrm{3x^2 + 6x^2 = 9x^2}\)
  • \(\mathrm{x}\) terms: \(\mathrm{-5x + 20x = 15x}\)
  • Constants: \(\mathrm{4 + (-13) = -9}\)

• Result: \(\mathrm{f(x) + g(x) = 9x^2 + 15x - 9}\)

3. INFER the factoring strategy

• To find a factor, look for what divides all terms
• Check if coefficients 9, 15, and -9 have a common divisor

4. SIMPLIFY by finding the GCD

• Factor each coefficient:
  • \(\mathrm{9 = 3 \times 3}\)
  • \(\mathrm{15 = 3 \times 5}\)
  • \(\mathrm{-9 = 3 \times (-3)}\)

• The GCD is 3 since it divides all coefficients

• Factor out 3: \(\mathrm{9x^2 + 15x - 9 = 3(3x^2 + 5x - 3)}\)

Answer: A (3)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making sign errors when combining like terms, especially with the constants \(\mathrm{4 + (-13)}\).

Students might calculate \(\mathrm{4 + (-13) = 17}\) instead of -9, or make errors with the x-coefficient \(\mathrm{(-5x + 20x)}\). This leads to an incorrect polynomial, making it impossible to identify the correct factor. This leads to confusion and guessing.

Second Most Common Error:

Missing INFER connection: Not recognizing that finding factors means looking for the GCD of coefficients.

Some students correctly find \(\mathrm{f(x) + g(x) = 9x^2 + 15x - 9}\) but then try to factor the entire quadratic expression rather than looking for common numerical factors. They might attempt to factor \(\mathrm{9x^2 + 15x - 9}\) as a product of binomials, which is more complex and unnecessary for this problem. This causes them to get stuck and guess.

The Bottom Line:

This problem tests whether students can systematically add polynomials and understand that finding "a factor" often means finding the simplest common factor of all terms, not necessarily factoring completely.