\((2\mathrm{x}^3 + 3\mathrm{x})(\mathrm{x}^3 - 2\mathrm{x})\) Which of the following is equivalent to the expression above?...

1. INFER the approach needed

• We have two polynomial expressions being multiplied: \((2\mathrm{x}^3 + 3\mathrm{x})(\mathrm{x}^3 - 2\mathrm{x})\)
• This requires using the distributive property to multiply each term in the first expression by each term in the second expression

2. SIMPLIFY by applying the distributive property

• Multiply each term in the first parentheses by each term in the second parentheses:
  • \(2\mathrm{x}^3 \cdot \mathrm{x}^3 = 2\mathrm{x}^6\)
  • \(2\mathrm{x}^3 \cdot (-2\mathrm{x}) = -4\mathrm{x}^4\)
  • \(3\mathrm{x} \cdot \mathrm{x}^3 = 3\mathrm{x}^4\)
  • \(3\mathrm{x} \cdot (-2\mathrm{x}) = -6\mathrm{x}^2\)
• This gives us: \(2\mathrm{x}^6 - 4\mathrm{x}^4 + 3\mathrm{x}^4 - 6\mathrm{x}^2\)

3. SIMPLIFY by combining like terms

• Identify like terms: \(-4\mathrm{x}^4\) and \(+3\mathrm{x}^4\) are both \(\mathrm{x}^4\) terms
• Combine them: \(-4\mathrm{x}^4 + 3\mathrm{x}^4 = -\mathrm{x}^4\)
• Final expression: \(2\mathrm{x}^6 - \mathrm{x}^4 - 6\mathrm{x}^2\)

Answer: C. \(2\mathrm{x}^6 - \mathrm{x}^4 - 6\mathrm{x}^2\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students misinterpret the operation and add the expressions instead of multiplying them.

They might think: "I have \((2\mathrm{x}^3 + 3\mathrm{x})\) and \((\mathrm{x}^3 - 2\mathrm{x})\), so I'll just add the like terms: \(2\mathrm{x}^3 + \mathrm{x}^3 = 3\mathrm{x}^3\) and \(3\mathrm{x} - 2\mathrm{x} = \mathrm{x}\), giving me \(3\mathrm{x}^3 + \mathrm{x}\)."

This leads them to select Choice B (\(3\mathrm{x}^3 + \mathrm{x}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly use the distributive property but make errors with exponent rules or arithmetic.

For example, they might incorrectly calculate \(2\mathrm{x}^3 \cdot \mathrm{x}^3 = 3\mathrm{x}^6\) instead of \(2\mathrm{x}^6\), or make sign errors when combining like terms. These computational mistakes often result in incorrect leading coefficients.

This may lead them to select Choice D (\(3\mathrm{x}^6 - \mathrm{x}^4 - 6\mathrm{x}^2\)) due to the wrong first term.

The Bottom Line:

This problem tests whether students can systematically apply the distributive property to polynomial multiplication while maintaining accuracy through multiple algebraic steps. The key is recognizing that parentheses indicate multiplication, not addition, and then executing the distribution carefully.