Question:Which of the following is equivalent to sqrt[3]{x^3 + 12x^2 + 48x + 64}, where x gt 0?\((\mathrm{x} + 4)^3\)\((\mathrm{x}...

1. INFER the solution strategy

• Looking at \(\sqrt[3]{\mathrm{x^3 + 12x^2 + 48x + 64}}\), the key insight is recognizing that the polynomial inside might be a perfect cube
• Strategy: Check if \(\mathrm{x^3 + 12x^2 + 48x + 64 = (x + a)^3}\) for some value a

2. INFER what value to test

• Since the leading term is \(\mathrm{x^3}\) and the constant is \(\mathrm{64 = 4^3}\), let's test if this equals \(\mathrm{(x + 4)^3}\)
• We need to verify using the perfect cube expansion: \(\mathrm{(x + a)^3 = x^3 + 3ax^2 + 3a^2x + a^3}\)

3. SIMPLIFY by checking each coefficient

• For \(\mathrm{(x + 4)^3 = x^3 + 3(4)x^2 + 3(4^2)x + 4^3}\)
• This gives us: \(\mathrm{x^3 + 12x^2 + 48x + 64}\) ✓
• Perfect match! So \(\mathrm{x^3 + 12x^2 + 48x + 64 = (x + 4)^3}\)

4. SIMPLIFY using the cube root property

• \(\sqrt[3]{\mathrm{x^3 + 12x^2 + 48x + 64}} = \sqrt[3]{\mathrm{(x + 4)^3}}\)
• Using \(\sqrt[3]{\mathrm{a^3}} = \mathrm{a}\): \(\sqrt[3]{\mathrm{(x + 4)^3}} = \mathrm{x + 4}\)
• Since \(\mathrm{x \gt 0}\), we have \(\mathrm{x + 4 \gt 0}\), so no absolute value concerns

Answer: E

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the perfect cube pattern and instead try to factor out common terms or use other inappropriate methods.

They might factor out small terms like x from some terms, getting stuck with expressions like \(\mathrm{x(x^2 + 12x + 48) + 64}\), which doesn't lead anywhere useful. Without recognizing the perfect cube structure, they can't make progress and end up guessing from the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students recognize they should test for a perfect cube but make calculation errors when verifying the factorization.

For example, they might test \(\mathrm{(x + 4)^3}\) but incorrectly calculate \(\mathrm{3(4^2) = 3(16)}\) as 24 instead of 48, concluding the factorization doesn't work. This leads them to abandon the correct approach and potentially select Choice D (\(\mathrm{(x + 4)^{1/3}}\)) thinking the cube root "partially cancels."

The Bottom Line:

This problem requires pattern recognition to identify perfect cubes hidden within seemingly complex polynomials. Success depends on systematically testing factorization rather than getting overwhelmed by the polynomial's appearance.