Which expression is equivalent to \(-3(\mathrm{x}^3 - 5)\)?

1. INFER the approach needed

• We have \(-3(x^3 - 5)\) and need to find an equivalent expression
• Since we have a number multiplied by a parenthetical expression, we should use the distributive property
• The distributive property states: \(a(b - c) = ab - ac\)

2. SIMPLIFY by distributing the -3

• First term: \((-3) \times (x^3) = -3x^3\)
• Second term: \((-3) \times (-5) = +15\)
  • Remember: negative × negative = positive
• Combine the results: \(-3x^3 + 15\)

Answer: C (\(-3x^3 + 15\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when distributing -3 to the second term.
They correctly get \(-3x^3\) for the first term, but then calculate \((-3) \times (-5) = -15\) instead of +15, forgetting that negative times negative equals positive.
This may lead them to select Choice A (\(-3x^3 - 15\)).

Second Most Common Error:

Incomplete SIMPLIFY reasoning: Students distribute -3 only to the first term and leave the second term unchanged.
They get \(-3x^3\) but don't multiply -3 by the -5, treating the expression as \(-3x^3 - 5\).
This may lead them to select Choice B (\(-3x^3 - 5\)).

The Bottom Line:

The key challenge is carefully applying the distributive property while tracking negative signs. Students must remember that distributing means multiplying the outside term by every term inside the parentheses, and they must apply sign rules correctly throughout.