Which of the following is equivalent to \((8\mathrm{x}^5 - 3\mathrm{x}^4) - (2\mathrm{x}^5 - 7\mathrm{x}^4)\)?

1. TRANSLATE the problem information

• Given: \((8x^5 - 3x^4) - (2x^5 - 7x^4)\)
• We need to subtract the second polynomial from the first

2. SIMPLIFY by distributing the subtraction

• When subtracting a polynomial, distribute the negative sign to each term:
• \((8x^5 - 3x^4) - (2x^5 - 7x^4) = 8x^5 - 3x^4 - 2x^5 + 7x^4\)
• Notice: \(-(2x^5 - 7x^4) = -2x^5 + 7x^4\)

3. SIMPLIFY by combining like terms

• Group terms with the same variable and exponent:
• \(x^5\) terms: \(8x^5 - 2x^5 = 6x^5\)
• \(x^4\) terms: \(-3x^4 + 7x^4 = 4x^4\)
• Final result: \(6x^5 + 4x^4\)

Answer: D (\(6x^5 + 4x^4\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when distributing the subtraction, particularly forgetting that \(-(-7x^4) = +7x^4\).

They might incorrectly write: \(8x^5 - 3x^4 - 2x^5 - 7x^4\), leading to:

• \(x^5\) terms: \(8x^5 - 2x^5 = 6x^5\) ✓
• \(x^4\) terms: \(-3x^4 - 7x^4 = -10x^4\) ✗

This leads them to select Choice E (\(6x^5 - 10x^4\)).

Second Most Common Error:

Poor SIMPLIFY reasoning: Students correctly distribute but make arithmetic errors when combining coefficients.

For example, when combining \(-3x^4 + 7x^4\), they might calculate incorrectly and get \(10x^4\) instead of \(4x^4\), or combine the \(x^5\) terms incorrectly to get \(10x^5\).

This may lead them to select Choice A (\(6x^5 + 10x^4\)) or Choice C (\(10x^5 + 4x^4\)).

The Bottom Line:

This problem tests careful attention to signs during polynomial subtraction. The key insight is remembering that subtracting a negative term makes it positive, and staying organized when combining like terms.