Which expression is equivalent to \((\mathrm{x}^2 - 7)^2 - (\mathrm{x} - 3)(\mathrm{x} + 3)\)?x^4 - 15x^2 + 40x^4 - 15x^2...

1. TRANSLATE the expression into recognizable patterns

• Given expression: \((\mathrm{x}^2 - 7)^2 - (\mathrm{x} - 3)(\mathrm{x} + 3)\)
• What this tells us: We have a perfect square pattern \((\mathrm{x}^2 - 7)^2\) and a difference of squares pattern \((\mathrm{x} - 3)(\mathrm{x} + 3)\)

2. SIMPLIFY the first term using perfect squares

• Apply \((\mathrm{a} - \mathrm{b})^2 = \mathrm{a}^2 - 2\mathrm{ab} + \mathrm{b}^2\) where \(\mathrm{a} = \mathrm{x}^2\) and \(\mathrm{b} = 7\):

\((\mathrm{x}^2 - 7)^2 = (\mathrm{x}^2)^2 - 2(\mathrm{x}^2)(7) + 7^2\)
\(= \mathrm{x}^4 - 14\mathrm{x}^2 + 49\)

3. SIMPLIFY the second term using difference of squares

• Apply \((\mathrm{a} - \mathrm{b})(\mathrm{a} + \mathrm{b}) = \mathrm{a}^2 - \mathrm{b}^2\) where \(\mathrm{a} = \mathrm{x}\) and \(\mathrm{b} = 3\):

\((\mathrm{x} - 3)(\mathrm{x} + 3) = \mathrm{x}^2 - 3^2\)
\(= \mathrm{x}^2 - 9\)

4. SIMPLIFY by performing the subtraction

• Substitute expanded forms: \((\mathrm{x}^4 - 14\mathrm{x}^2 + 49) - (\mathrm{x}^2 - 9)\)
• Distribute the negative sign: \(\mathrm{x}^4 - 14\mathrm{x}^2 + 49 - \mathrm{x}^2 + 9\)
• Combine like terms: \(\mathrm{x}^4 + (-14\mathrm{x}^2 - \mathrm{x}^2) + (49 + 9)\)
  \(= \mathrm{x}^4 - 15\mathrm{x}^2 + 58\)

Answer: B. \(\mathrm{x}^4 - 15\mathrm{x}^2 + 58\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when distributing the negative sign in the subtraction step.

They might write: \((\mathrm{x}^4 - 14\mathrm{x}^2 + 49) - (\mathrm{x}^2 - 9) = \mathrm{x}^4 - 14\mathrm{x}^2 + 49 - \mathrm{x}^2 - 9\)

This gives them \(\mathrm{x}^4 - 15\mathrm{x}^2 + 40\) instead of the correct \(\mathrm{x}^4 - 15\mathrm{x}^2 + 58\).

This may lead them to select Choice A (\(\mathrm{x}^4 - 15\mathrm{x}^2 + 40\)).

Second Most Common Error:

Missing conceptual knowledge: Students who don't recognize the difference of squares pattern might try to expand \((\mathrm{x} - 3)(\mathrm{x} + 3)\) using FOIL method and make computational errors.

They might get \((\mathrm{x} - 3)(\mathrm{x} + 3) = \mathrm{x}^2 + 3\mathrm{x} - 3\mathrm{x} - 9 = \mathrm{x}^2 - 9\) correctly, but the extra steps increase chances of later calculation mistakes, leading to confusion and guessing.

The Bottom Line:

This problem tests pattern recognition for special polynomial products and careful attention to signs during subtraction. Students need to both identify the correct formulas to apply and execute the algebra precisely.