The given expression \((5\mathrm{x}^3 - 3) - (-4\mathrm{x}^3 + 8)\) is equivalent to bx^3 - 11, where b is a...

1. TRANSLATE the problem information

• Given: \((5\mathrm{x}^3 - 3) - (-4\mathrm{x}^3 + 8)\) is equivalent to \(\mathrm{b}\mathrm{x}^3 - 11\)
• Find: The value of constant b

2. SIMPLIFY the left side of the equation

• Distribute the negative sign through the second parentheses:
  • \((5\mathrm{x}^3 - 3) - (-4\mathrm{x}^3 + 8)\) becomes \((5\mathrm{x}^3 - 3) + (4\mathrm{x}^3 - 8)\)
  • Remember: Subtracting a negative is the same as adding a positive
• Combine like terms:
  • \(5\mathrm{x}^3 - 3 + 4\mathrm{x}^3 - 8\)
  • Group \(\mathrm{x}^3\) terms: \((5\mathrm{x}^3 + 4\mathrm{x}^3) = 9\mathrm{x}^3\)
  • Group constants: \((-3 - 8) = -11\)
  • Result: \(9\mathrm{x}^3 - 11\)

3. INFER the relationship between expressions

• Since \((5\mathrm{x}^3 - 3) - (-4\mathrm{x}^3 + 8) = \mathrm{b}\mathrm{x}^3 - 11\)
• And we simplified the left side to \(9\mathrm{x}^3 - 11\)
• We have: \(9\mathrm{x}^3 - 11 = \mathrm{b}\mathrm{x}^3 - 11\)

4. INFER the coefficient comparison

• For two expressions to be equivalent, their corresponding coefficients must be equal
• Coefficient of \(\mathrm{x}^3\) on left side: 9
• Coefficient of \(\mathrm{x}^3\) on right side: b
• Therefore: \(\mathrm{b} = 9\)

Answer: 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign errors when distributing the negative through parentheses

Students often write: \((5\mathrm{x}^3 - 3) - (-4\mathrm{x}^3 + 8) = 5\mathrm{x}^3 - 3 - 4\mathrm{x}^3 + 8\)

This incorrect distribution leads to:

• Combining: \((5\mathrm{x}^3 - 4\mathrm{x}^3) + (-3 + 8) = \mathrm{x}^3 + 5\)
• Final answer: \(\mathrm{b} = 1\)

This fundamental algebraic mistake stems from not carefully handling the "minus a negative" situation.

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic errors when combining constants

Students correctly distribute to get \(5\mathrm{x}^3 - 3 + 4\mathrm{x}^3 - 8\), but then make calculation errors:

• Incorrectly compute \(-3 - 8 = -5\) instead of \(-11\)
• This leads to \(9\mathrm{x}^3 - 5\), making them think the problem setup is wrong

The Bottom Line:

This problem tests whether students can systematically handle negative signs and combine like terms without losing track of signs—a fundamental algebraic skill that requires careful, step-by-step work.