If A = 3x^3 + 2x^2 - 4 and A + B = 7x^3 - 3x^2 + x + 1,...

1. INFER the strategy needed

• Given information:
  • \(\mathrm{A = 3x³ + 2x² - 4}\)
  • \(\mathrm{A + B = 7x³ - 3x² + x + 1}\)
  • Need to find B

• Key insight: Since we know A + B and we know A, we can find B by rearranging to \(\mathrm{B = (A + B) - A}\)

2. SIMPLIFY by setting up the subtraction

• Substitute the known expressions:
  \(\mathrm{B = (7x³ - 3x² + x + 1) - (3x³ + 2x² - 4)}\)

• The parentheses around the second polynomial are crucial because we need to subtract the entire expression

3. SIMPLIFY by distributing the negative sign

• \(\mathrm{B = 7x³ - 3x² + x + 1 - 3x³ - 2x² + 4}\)

• Every term in the second polynomial changes sign when we remove the parentheses

4. SIMPLIFY by combining like terms

• Group terms by degree:
  • x³ terms: \(\mathrm{7x³ - 3x³ = 4x³}\)
  • x² terms: \(\mathrm{-3x² - 2x² = -5x²}\)
  • x terms: x (only one x term)
  • Constants: \(\mathrm{1 + 4 = 5}\)

• \(\mathrm{B = 4x³ - 5x² + x + 5}\)

Answer: C (\(\mathrm{4x³ - 5x² + x + 5}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly distribute the negative sign when subtracting the polynomial A.

Instead of \(\mathrm{B = 7x³ - 3x² + x + 1 - 3x³ - 2x² + 4}\), they might write:
\(\mathrm{B = 7x³ - 3x² + x + 1 - 3x³ + 2x² - 4}\)

This leads to combining like terms incorrectly:

• x² terms: \(\mathrm{-3x² + 2x² = -x²}\) (instead of \(\mathrm{-5x²}\))
• Constants: \(\mathrm{1 - 4 = -3}\) (instead of +5)

This may lead them to select Choice A (\(\mathrm{4x³ - x² + x + 5}\)) or Choice B (\(\mathrm{4x³ - 5x² + x - 3}\)).

Second Most Common Error:

Missing INFER insight: Students might try to add A and B directly or become confused about how to isolate B from the given equation.

Without recognizing that \(\mathrm{B = (A + B) - A}\), they get stuck trying other approaches that don't lead anywhere productive. This leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can strategically rearrange polynomial equations and execute polynomial subtraction with careful attention to signs—two skills that build on each other sequentially.