Which expression is equivalent to \((8\mathrm{x}^3 + 8) - (\mathrm{x}^3 - 2)\)?

1. INFER what polynomial subtraction means

• When we subtract one polynomial from another, we must distribute the negative sign to every term in the subtracted polynomial
• The expression \((8\mathrm{x}^3 + 8) - (\mathrm{x}^3 - 2)\) means we subtract each term of \((\mathrm{x}^3 - 2)\)

2. SIMPLIFY by distributing the negative sign

• \((8\mathrm{x}^3 + 8) - (\mathrm{x}^3 - 2)\)
• Distribute the negative: \(8\mathrm{x}^3 + 8 - \mathrm{x}^3 - (-2)\)
• Remember: subtracting a negative number means adding: \(8\mathrm{x}^3 + 8 - \mathrm{x}^3 + 2\)

3. SIMPLIFY by combining like terms

• Group the x³ terms: \(8\mathrm{x}^3 - \mathrm{x}^3 = 7\mathrm{x}^3\)
• Group the constant terms: \(8 + 2 = 10\)
• Final result: \(7\mathrm{x}^3 + 10\)

Answer: B. \(7\mathrm{x}^3 + 10\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students often struggle with distributing the negative sign correctly, especially when dealing with the term \((-2)\).

Many students write: \((8\mathrm{x}^3 + 8) - (\mathrm{x}^3 - 2) = 8\mathrm{x}^3 + 8 - \mathrm{x}^3 - 2\)

They forget that subtracting \((-2)\) actually means adding \((+2)\). This gives them \(8\mathrm{x}^3 + 8 - \mathrm{x}^3 - 2 = 7\mathrm{x}^3 + 6\).

This may lead them to select Choice D (\(7\mathrm{x}^3 + 6\)).

Second Most Common Error:

Incomplete SIMPLIFY process: Some students correctly distribute the negative sign but fail to properly combine the x³ terms.

They might write: \(8\mathrm{x}^3 + 8 - \mathrm{x}^3 + 2 = 8\mathrm{x}^3 + 10\) (forgetting to subtract x³ from 8x³).

This may lead them to select Choice C (\(8\mathrm{x}^3 + 10\)).

The Bottom Line:

Polynomial subtraction requires careful attention to sign changes and systematic combining of like terms. The key insight is that subtracting a polynomial means adding its opposite, which changes the sign of every term in the subtracted expression.