Let a = 2x + 3 and b = 5 - x. What is the coefficient of x in the...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{a = 2x + 3}\)
  • \(\mathrm{b = 5 - x}\)
  • Need coefficient of x in \(\mathrm{(a - 2b)(a + b) + 3a}\)

2. SIMPLIFY by finding a - 2b

• Substitute and distribute carefully:

\(\mathrm{a - 2b = (2x + 3) - 2(5 - x)}\)

\(\mathrm{= 2x + 3 - 10 + 2x}\)

\(\mathrm{= 4x - 7}\)

3. SIMPLIFY by finding a + b

• Add the expressions:

\(\mathrm{a + b = (2x + 3) + (5 - x)}\)

\(\mathrm{= x + 8}\)

4. SIMPLIFY by expanding (a - 2b)(a + b)

• Use FOIL method:

\(\mathrm{(4x - 7)(x + 8) = 4x^2 + 32x - 7x - 56}\)

\(\mathrm{= 4x^2 + 25x - 56}\)

5. SIMPLIFY by finding 3a

• Distribute:

\(\mathrm{3a = 3(2x + 3) = 6x + 9}\)

6. SIMPLIFY by combining all terms

• Add the results:

\(\mathrm{(4x^2 + 25x - 56) + (6x + 9) = 4x^2 + 31x - 47}\)

The coefficient of x is 31.

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign errors when distributing \(\mathrm{-2(5 - x)}\)

Students often write: \(\mathrm{a - 2b = (2x + 3) - 2(5 - x) = 2x + 3 - 10 - 2x = -7}\)

They forget that -2 times (-x) gives +2x, not -2x. This leads to \(\mathrm{a - 2b = -7}\) instead of \(\mathrm{4x - 7}\), making the final coefficient much smaller.

This may lead them to select Choice A (7) or get confused and guess.

Second Most Common Error:

Incomplete SIMPLIFY process: Forgetting to add 3a at the end

Students correctly expand \(\mathrm{(a - 2b)(a + b)}\) to get \(\mathrm{4x^2 + 25x - 56}\), identify the coefficient as 25, but forget the problem asks for \(\mathrm{(a - 2b)(a + b) + 3a}\).

This may lead them to select Choice C (25).

The Bottom Line:

This problem tests sustained algebraic manipulation through multiple steps. Success requires careful attention to signs and completing all required operations.