If u = 5x - 2 and w = 2x + 3, which of the following is equivalent to uw...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{u = 5x - 2}\)
  • \(\mathrm{w = 2x + 3}\)
  • Need to find: \(\mathrm{uw + 3u - 2w}\)

• What this tells us: We need to substitute these expressions for u and w everywhere they appear.

2. TRANSLATE by substituting the expressions

• Replace each variable with its definition:
  • uw becomes \(\mathrm{(5x - 2)(2x + 3)}\)
  • 3u becomes \(\mathrm{3(5x - 2)}\)
  • 2w becomes \(\mathrm{2(2x + 3)}\)
• Our expression becomes: \(\mathrm{(5x - 2)(2x + 3) + 3(5x - 2) - 2(2x + 3)}\)

3. SIMPLIFY by expanding the binomial product first

• Expand \(\mathrm{(5x - 2)(2x + 3)}\) using FOIL:
  • First: \(\mathrm{5x \times 2x = 10x^2}\)
  • Outer: \(\mathrm{5x \times 3 = 15x}\)
  • Inner: \(\mathrm{-2 \times 2x = -4x}\)
  • Last: \(\mathrm{-2 \times 3 = -6}\)
• Result: \(\mathrm{10x^2 + 15x - 4x - 6 = 10x^2 + 11x - 6}\)

4. SIMPLIFY the remaining terms using distributive property

• \(\mathrm{3(5x - 2) = 15x - 6}\)
• \(\mathrm{-2(2x + 3) = -4x - 6}\)

5. SIMPLIFY by combining all terms

• Write out all terms: \(\mathrm{10x^2 + 11x - 6 + 15x - 6 - 4x - 6}\)
• Group like terms: \(\mathrm{10x^2 + (11x + 15x - 4x) + (-6 - 6 - 6)}\)
• Combine: \(\mathrm{10x^2 + 22x - 18}\)

Answer: C. \(\mathrm{10x^2 + 22x - 18}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making sign errors when expanding \(\mathrm{(5x - 2)(2x + 3)}\), particularly with the negative terms. Students might get \(\mathrm{10x^2 + 15x + 4x - 6}\) instead of \(\mathrm{10x^2 + 15x - 4x - 6}\), leading to incorrect like-term combination.

This typically results in \(\mathrm{10x^2 + 25x - 12}\) after combining all terms, leading them to select Choice D \(\mathrm{(10x^2 + 25x - 6)}\) or getting confused between similar-looking answer choices.

Second Most Common Error:

Incomplete SIMPLIFY process: Students correctly expand the binomial but make arithmetic errors when combining like terms, especially when dealing with multiple negative constants \(\mathrm{(-6 - 6 - 6 = -18)}\). They might incorrectly calculate the constant term or lose track of x-coefficients.

This leads to selecting Choice A \(\mathrm{(10x^2 + 7x - 18)}\) or Choice B \(\mathrm{(10x^2 + 17x - 12)}\) depending on which terms they miscombine.

The Bottom Line:

This problem requires careful attention to signs and systematic organization when handling multiple algebraic terms simultaneously. Success depends on methodical expansion and meticulous like-term combination.