\(\mathrm{f(x) = x^3 - 9x}\)\(\mathrm{g(x) = x^2 - 2x - 3}\)Which of the following expressions is equivalent to \(\mathrm{\frac{f(x)}{g(x)}}\), for...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = x^3 - 9x}\)
  • \(\mathrm{g(x) = x^2 - 2x - 3}\)
  • Find \(\mathrm{f(x)/g(x)}\) for \(\mathrm{x \gt 3}\)

2. INFER the solution strategy

• To simplify \(\mathrm{f(x)/g(x)}\), we need to factor both polynomials first
• Look for common factors that might cancel out
• The constraint \(\mathrm{x \gt 3}\) will be important for justifying any cancellations

3. SIMPLIFY f(x) by factoring

• \(\mathrm{f(x) = x^3 - 9x}\)
• Factor out the common x: \(\mathrm{f(x) = x(x^2 - 9)}\)
• Recognize \(\mathrm{x^2 - 9}\) as difference of squares: \(\mathrm{f(x) = x(x + 3)(x - 3)}\)

4. SIMPLIFY g(x) by factoring

• \(\mathrm{g(x) = x^2 - 2x - 3}\)
• Find two numbers that multiply to -3 and add to -2: those are -3 and +1
• \(\mathrm{g(x) = (x - 3)(x + 1)}\)

5. SIMPLIFY the rational expression

• \(\mathrm{f(x)/g(x) = \frac{x(x + 3)(x - 3)}{(x - 3)(x + 1)}}\)
• Notice the common factor \(\mathrm{(x - 3)}\) in numerator and denominator

6. APPLY CONSTRAINTS to cancel safely

• Since \(\mathrm{x \gt 3}\), we know \(\mathrm{(x - 3) ≠ 0}\)
• Therefore we can cancel: \(\mathrm{f(x)/g(x) = \frac{x(x + 3)}{(x + 1)}}\)

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

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students struggle with factoring the polynomials completely, especially the cubic \(\mathrm{f(x) = x^3 - 9x}\). They might factor out the x but miss that \(\mathrm{x^2 - 9}\) is a difference of squares, leaving \(\mathrm{f(x) = x(x^2 - 9)}\) and getting stuck.

This leads to confusion and guessing since they can't proceed without proper factoring.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students successfully factor both expressions and see the common \(\mathrm{(x - 3)}\) factor, but worry about canceling it without considering the given constraint \(\mathrm{x \gt 3}\). This hesitation might lead them to leave the expression unsimplified or choose a more complex-looking answer.

This may lead them to select Choice C (\(\mathrm{\frac{x(x-3)}{(x+1)}}\)) by incorrectly thinking they can't cancel the common factor.

The Bottom Line:

This problem tests whether students can systematically factor polynomials and use domain constraints to justify algebraic operations. Success requires both technical factoring skills and strategic thinking about when cancellations are valid.