Question:\(\mathrm{h(x) = 2x^3 + 8x^2 - 10x}\)\(\mathrm{k(x) = x^2 + 5x - 6}\)Which of the following expressions is equivalent to...

1. INFER the solution strategy

• To divide polynomials, we need to factor both the numerator and denominator first
• Look for common factors that can be canceled out
• The condition "\(\mathrm{x \gt 1}\)" suggests we'll encounter a factor we can cancel

2. SIMPLIFY the numerator h(x)

• Start with \(\mathrm{h(x) = 2x^3 + 8x^2 - 10x}\)
• Factor out the greatest common factor: \(\mathrm{h(x) = 2x(x^2 + 4x - 5)}\)
• Factor the quadratic \(\mathrm{x^2 + 4x - 5}\):
  • Need two numbers that multiply to -5 and add to 4
  • These are 5 and -1
  • So \(\mathrm{x^2 + 4x - 5 = (x + 5)(x - 1)}\)
• Complete factorization: \(\mathrm{h(x) = 2x(x + 5)(x - 1)}\)

3. SIMPLIFY the denominator k(x)

• Start with \(\mathrm{k(x) = x^2 + 5x - 6}\)
• Factor the quadratic:
  • Need two numbers that multiply to -6 and add to 5
  • These are 6 and -1
  • So \(\mathrm{k(x) = (x + 6)(x - 1)}\)

4. SIMPLIFY the quotient

• Write out the division: \(\mathrm{h(x)/k(x) = \frac{2x(x + 5)(x - 1)}{(x + 6)(x - 1)}}\)
• Notice the common factor \(\mathrm{(x - 1)}\) in both numerator and denominator

5. APPLY CONSTRAINTS to cancel factors

• Since \(\mathrm{x \gt 1}\), we know \(\mathrm{x \neq 1}\), so \(\mathrm{(x - 1) \neq 0}\)
• This allows us to cancel the common factor: \(\mathrm{h(x)/k(x) = \frac{2x(x + 5)}{(x + 6)}}\)

Answer: (D) \(\mathrm{\frac{2x(x + 5)}{(x + 6)}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students struggle with factoring the polynomials completely or make errors in the factoring process.

For example, they might incorrectly factor \(\mathrm{x^2 + 4x - 5}\) as \(\mathrm{(x + 4)(x - 1)}\) instead of \(\mathrm{(x + 5)(x - 1)}\), or fail to factor out the \(\mathrm{2x}\) from the numerator initially. This leads to an incorrect setup that doesn't allow for proper cancellation, causing them to select Choice (A) \(\mathrm{\frac{2x}{(x + 6)}}\) or get confused and guess.

Second Most Common Error:

Poor INFER reasoning: Students attempt to divide the polynomials directly without factoring first, or they factor but don't recognize they can cancel common factors.

They might try polynomial long division instead of the more efficient factoring approach, or see the factored form but miss that \(\mathrm{(x - 1)}\) appears in both parts and can be canceled. This leads to either getting stuck in complex calculations or arriving at an unfactored expression, potentially selecting Choice (B) \(\mathrm{\frac{2(x + 5)}{(x + 6)}}\).

The Bottom Line:

This problem tests whether students can systematically factor polynomials and understand when and how to cancel common factors in rational expressions. The key insight is recognizing that factoring enables cancellation, making a complex-looking division become much simpler.