If \(2(3\mathrm{t}^3 - \mathrm{t}^2 + 4) - (5\mathrm{t}^3 + 3\mathrm{t} - 6) + \mathrm{t}^2\) is simplified, which of the following...

1. SIMPLIFY by distributing coefficients

• First, distribute the 2 across the first parentheses:
  • \(2(3\mathrm{t}^3 − \mathrm{t}^2 + 4) = 6\mathrm{t}^3 − 2\mathrm{t}^2 + 8\)
• Next, distribute the negative sign across the second parentheses:
  • \(−(5\mathrm{t}^3 + 3\mathrm{t} − 6) = −5\mathrm{t}^3 − 3\mathrm{t} + 6\)
  • Remember: the negative flips ALL signs inside

2. SIMPLIFY by rewriting the complete expression

• Now we have: \(6\mathrm{t}^3 − 2\mathrm{t}^2 + 8 − 5\mathrm{t}^3 − 3\mathrm{t} + 6 + \mathrm{t}^2\)

3. SIMPLIFY by combining like terms systematically

• Group terms by degree (highest to lowest):
  • Cubic terms \((\mathrm{t}^3)\): \(6\mathrm{t}^3 − 5\mathrm{t}^3 = \mathrm{t}^3\)
  • Quadratic terms \((\mathrm{t}^2)\): \(−2\mathrm{t}^2 + \mathrm{t}^2 = −\mathrm{t}^2\)
  • Linear terms \((\mathrm{t})\): \(−3\mathrm{t}\)
  • Constant terms: \(8 + 6 = 14\)
• Final result: \(\mathrm{t}^3 − \mathrm{t}^2 − 3\mathrm{t} + 14\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution with negative distribution: Students often struggle with distributing the negative sign correctly, forgetting to change the sign of every term inside the parentheses.

For example, they might write \(−(5\mathrm{t}^3 + 3\mathrm{t} − 6) = −5\mathrm{t}^3 + 3\mathrm{t} − 6\) (forgetting to flip the sign of the \(3\mathrm{t}\) term). This error would give them a final result of \(\mathrm{t}^3 − \mathrm{t}^2 + 3\mathrm{t} + 14\), leading them to select Choice C.

Second Most Common Error:

Poor SIMPLIFY organization when combining like terms: Students may incorrectly combine unlike terms or make arithmetic errors when adding/subtracting coefficients.

A common mistake is miscalculating the constant terms \((8 + 6)\) or the quadratic terms \((−2\mathrm{t}^2 + \mathrm{t}^2)\), which could lead them to Choice B \((\mathrm{t}^3 − \mathrm{t}^2 − 3\mathrm{t} + 10)\) or Choice D \((\mathrm{t}^3 − \mathrm{t}^2 − 3\mathrm{t} + 2)\).

The Bottom Line:

This problem tests careful execution of fundamental algebraic operations. Success requires methodical attention to signs during distribution and systematic organization when combining like terms.