Question:Which of the following is equivalent to the expression y^6 + y^3 - 12?\(\mathrm{(y^3 + 1)(y^3 - 12)}\)\(\mathrm{(y^3 + 2)(y^3...

1. INFER the strategic approach

• Given expression: \(\mathrm{y^6 + y^3 - 12}\)
• Key insight: This looks like a quadratic if we think of \(\mathrm{y^3}\) as a single unit
• Strategy: Use substitution to convert this to a standard quadratic form

2. TRANSLATE using substitution

• Let \(\mathrm{u = y^3}\)
• The expression becomes: \(\mathrm{u^2 + u - 12}\)
• Now we have a standard quadratic to factor

3. SIMPLIFY by factoring the quadratic

• Need two numbers that multiply to \(\mathrm{-12}\) and add to \(\mathrm{+1}\)
• Test factor pairs systematically:
  • \(\mathrm{1}\) and \(\mathrm{-12}\): sum = \(\mathrm{-11}\) ❌
  • \(\mathrm{2}\) and \(\mathrm{-6}\): sum = \(\mathrm{-4}\) ❌
  • \(\mathrm{3}\) and \(\mathrm{-4}\): sum = \(\mathrm{-1}\) ❌
  • \(\mathrm{4}\) and \(\mathrm{-3}\): sum = \(\mathrm{+1}\) ✓
• Therefore: \(\mathrm{u^2 + u - 12 = (u + 4)(u - 3)}\)

4. TRANSLATE back to original variable

• Substitute \(\mathrm{y^3}\) for \(\mathrm{u}\): \(\mathrm{(y^3 + 4)(y^3 - 3)}\)
• This matches answer choice D

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students find incorrect factor pairs for the quadratic.

Many students rush through finding factors and select pairs like \(\mathrm{3}\) and \(\mathrm{-4}\) (which give sum = \(\mathrm{-1}\), not \(\mathrm{+1}\)) or \(\mathrm{2}\) and \(\mathrm{-6}\) (which give sum = \(\mathrm{-4}\), not \(\mathrm{+1}\)). When they use \(\mathrm{3}\) and \(\mathrm{-4}\), they get \(\mathrm{(y^3 + 3)(y^3 - 4)}\), leading them to select Choice C.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize the substitution opportunity.

Some students attempt to factor \(\mathrm{y^6 + y^3 - 12}\) directly without seeing the quadratic pattern. They might try to group terms or look for common factors that don't exist, leading to confusion and random guessing among the answer choices.

The Bottom Line:

This problem tests whether students can recognize hidden patterns in algebraic expressions. The key breakthrough is seeing that expressions with terms like \(\mathrm{y^6}\) and \(\mathrm{y^3}\) often factor like quadratics when you think of the repeated term (\(\mathrm{y^3}\)) as a single unit.