Question:Consider the expression (8x^3 + 16)/4. Which of the following expressions is equivalent to this expression?2x^3 + 4\(2(\mathrm{x}^3 + 4)\)x^3...

1. INFER the approach strategy

Given: \(\frac{8\mathrm{x}^3 + 16}{4}\)

You have two main approaches:

• Distribute the division to each term in the numerator
• Factor the numerator first, then simplify

Both work equally well - let's try the distribution method first.

2. SIMPLIFY by distributing division

Apply the distributive property of division:

\(\frac{8\mathrm{x}^3 + 16}{4} = \frac{8\mathrm{x}^3}{4} + \frac{16}{4}\)

Now divide each term:

• \(\frac{8\mathrm{x}^3}{4} = 2\mathrm{x}^3\)
• \(\frac{16}{4} = 4\)

Result: \(2\mathrm{x}^3 + 4\)

3. Verify using the factoring approach

SIMPLIFY by factoring first:

• Factor out 8 from numerator: \((8\mathrm{x}^3 + 16) = 8(\mathrm{x}^3 + 2)\)
• Substitute: \(\frac{8(\mathrm{x}^3 + 2)}{4} = 2(\mathrm{x}^3 + 2) = 2\mathrm{x}^3 + 4\)

Both methods give the same result!

Answer: A \((2\mathrm{x}^3 + 4)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students fail to distribute division to both terms in the numerator.

They correctly compute \(\frac{8\mathrm{x}^3}{4} = 2\mathrm{x}^3\) but forget to divide the constant term, writing:

\(\frac{8\mathrm{x}^3 + 16}{4} = 2\mathrm{x}^3 + 16\)

This leads them to select Choice D \((8\mathrm{x}^3 + 4)\) - wait, that doesn't match. Let me reconsider...

Actually, if they write \(2\mathrm{x}^3 + 16\), none of the choices match exactly. More likely they make the error of not dividing the \(8\mathrm{x}^3\) term properly, getting \(\frac{\mathrm{x}^3 + 16}{4}\), then \(\mathrm{x}^3 + 4\), which still doesn't match the choices perfectly.

The most realistic error is incomplete distribution: they might correctly divide \(\frac{16}{4} = 4\), but incorrectly think \(\frac{8\mathrm{x}^3}{4} = 8\mathrm{x}^3\), leading to Choice D \((8\mathrm{x}^3 + 4)\).

Second Most Common Error:

Conceptual confusion about factoring: Students who choose the factoring approach might incorrectly think \((8\mathrm{x}^3 + 16)\) factors as \(8(\mathrm{x}^3 + 4)\) instead of \(8(\mathrm{x}^3 + 2)\).

This leads them to compute \(\frac{8(\mathrm{x}^3 + 4)}{4} = 2(\mathrm{x}^3 + 4)\), selecting Choice B \([2(\mathrm{x}^3 + 4)]\) without expanding to check that this equals \(2\mathrm{x}^3 + 8\), not \(2\mathrm{x}^3 + 4\).

The Bottom Line:

This problem tests careful execution of basic algebraic operations. The key insight is recognizing that division must be applied to ALL terms in the numerator, not just some of them.