Let \(\mathrm{h(x) = x^2 + cx}\) and \(\mathrm{j(x) = 8x - 24}\). For all nonzero x, \(\mathrm{\frac{h(x) \cdot j(x)}{x} =...

1. TRANSLATE the given information

• Given information:
  • \(\mathrm{h(x) = x^2 + cx}\)
  • \(\mathrm{j(x) = 8x - 24}\)
  • \(\frac{\mathrm{h(x) \cdot j(x)}}{\mathrm{x}} = \mathrm{8x^2 - 22x - 6}\)
• What this tells us: We need to find the value of c that makes this equation true for all nonzero x.

2. INFER the solution approach

• Since we have a functional equation involving the product \(\mathrm{h(x) \cdot j(x)}\), we should:
  • First compute the product \(\mathrm{h(x) \cdot j(x)}\)
  • Then divide by x to get the left side
  • Compare with the right side to find c

3. SIMPLIFY by expanding the product

• Compute \(\mathrm{h(x) \cdot j(x)}\):
  \(\mathrm{(x^2 + cx)(8x - 24)}\)

• Use distributive property:
  \(\mathrm{= x^2(8x - 24) + cx(8x - 24)}\)
  \(\mathrm{= 8x^3 - 24x^2 + 8cx^2 - 24cx}\)
  \(\mathrm{= 8x^3 + (8c - 24)x^2 - 24cx}\)

4. SIMPLIFY by dividing by x

• \(\frac{\mathrm{h(x) \cdot j(x)}}{\mathrm{x}} = \frac{\mathrm{8x^3 + (8c - 24)x^2 - 24cx}}{\mathrm{x}}\)
• \(\mathrm{= 8x^2 + (8c - 24)x - 24c}\)

5. INFER using coefficient comparison

• We now have: \(\mathrm{8x^2 + (8c - 24)x - 24c = 8x^2 - 22x - 6}\)

• For polynomials to be equal, corresponding coefficients must be equal:
  • \(\mathrm{x^2}\) coefficient: \(\mathrm{8 = 8}\) ✓ (automatically satisfied)
  • x coefficient: \(\mathrm{8c - 24 = -22}\)
  • constant term: \(\mathrm{-24c = -6}\)

6. SIMPLIFY to solve for c

• From x coefficient: \(\mathrm{8c - 24 = -22}\)
  \(\mathrm{8c = 2}\)
  \(\mathrm{c = \frac{1}{4}}\)

• Check with constant term: \(\mathrm{-24c = -24(\frac{1}{4}) = -6}\) ✓

Answer: D. 1/4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when expanding \(\mathrm{(x^2 + cx)(8x - 24)}\), often forgetting to distribute properly or making sign errors in the expansion.

For example, they might get \(\mathrm{8x^3 - 24x^2 + 8cx^2 + 24cx}\) instead of \(\mathrm{8x^3 - 24x^2 + 8cx^2 - 24cx}\), leading to incorrect coefficient equations and wrong values for c.

This may lead them to select Choice A (-3/2) or get confused and guess.

Second Most Common Error:

Poor INFER reasoning: Students solve only one coefficient equation instead of checking their answer against all coefficients. They might solve \(\mathrm{8c - 24 = -22}\) to get \(\mathrm{c = \frac{1}{4}}\), but fail to verify that \(\mathrm{-24c = -6}\) when \(\mathrm{c = \frac{1}{4}}\).

Without this verification step, they might second-guess their correct answer or make computational errors that lead to selecting Choice B (-1/4) or Choice C (1/6).

The Bottom Line:

This problem requires careful polynomial algebra and systematic coefficient matching. The key insight is that when two polynomials are equal for all values of x, their corresponding coefficients must be equal - and checking multiple coefficient equations serves as a valuable verification step.