Question: Which expression is equivalent to 14n^3 + 28n^2 + 7n?\(7\mathrm{n}(2\mathrm{n}^2 + 4\mathrm{n} + 1)\)\(7\mathrm{n}(2\mathrm{n}^2 + 4\mathrm{...

1. INFER the problem type and strategy

• This is asking for an equivalent expression, which means we need to factor
• Strategy: Find the Greatest Common Factor (GCF) and factor it out
• All terms contain both a coefficient and the variable n, so we can factor out both

2. SIMPLIFY by finding the GCF of coefficients

• Coefficients are 14, 28, and 7
• Find factors: \(\mathrm{14 = 2 \times 7}\), \(\mathrm{28 = 4 \times 7}\), \(\mathrm{7 = 1 \times 7}\)
• GCF of coefficients = 7

3. SIMPLIFY by finding the GCF of variables

• Variables are \(\mathrm{n^3}\), \(\mathrm{n^2}\), and \(\mathrm{n}\)
• The GCF is the lowest power: \(\mathrm{n^1 = n}\)
• Overall GCF = \(\mathrm{7n}\)

4. SIMPLIFY by factoring out the GCF

• Divide each term by \(\mathrm{7n}\):

• \(\mathrm{14n^3 \div 7n = 2n^2}\)
• \(\mathrm{28n^2 \div 7n = 4n}\)
• \(\mathrm{7n \div 7n = 1}\)

• Result: \(\mathrm{7n(2n^2 + 4n + 1)}\)

5. Verify the answer

• Expand: \(\mathrm{7n(2n^2 + 4n + 1) = 14n^3 + 28n^2 + 7n}\) ✓

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students find the wrong GCF, often factoring out just 7 instead of \(\mathrm{7n}\), or factoring out \(\mathrm{14n}\) instead of \(\mathrm{7n}\).

For example, factoring out just 7 gives: \(\mathrm{7(2n^3 + 4n^2 + n)}\), which matches none of the choices. Factoring out \(\mathrm{14n}\) gives: \(\mathrm{14n(n^2 + 2n + 1/2)}\), but this creates a fraction that doesn't match any choice either. This leads to confusion and guessing.

Second Most Common Error:

Incomplete SIMPLIFY process: Students correctly identify \(\mathrm{7n}\) as the GCF but make arithmetic errors when dividing terms, such as getting \(\mathrm{28n^2 \div 7n = 3n}\) instead of \(\mathrm{4n}\).

This type of calculation error might lead them to select Choice B (\(\mathrm{7n(2n^2 + 4n)}\)) if they forget the constant term entirely, thinking they've factored out everything.

The Bottom Line:

This problem tests systematic factoring skills - students must correctly find the GCF of both coefficients and variables, then execute the division accurately for each term. The key insight is recognizing that every term contains both the number 7 and the variable n as common factors.