5x + 15 Which of the following is equivalent to the given expression?...

1. INFER the strategy needed

• The problem asks for an equivalent expression
• Looking at \(\mathrm{5x + 15}\), I should check if there's a common factor I can factor out
• Both terms contain the number 5, so factoring is the right approach

2. SIMPLIFY by factoring out the common factor

• Identify the common factor: Both \(\mathrm{5x}\) and \(\mathrm{15}\) are divisible by 5
• Factor out 5: \(\mathrm{5x + 15 = 5(? + ?)}\)
• Divide each term by 5:
  • \(\mathrm{5x ÷ 5 = x}\)
  • \(\mathrm{15 ÷ 5 = 3}\)
• Result: \(\mathrm{5x + 15 = 5(x + 3)}\)

3. Verify the answer

• Check by distributing: \(\mathrm{5(x + 3) = 5x + 15}\) ✓
• This matches the original expression

Answer: A. \(\mathrm{5(x + 3)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when dividing 15 by 5, incorrectly getting 10 instead of 3.

They might think: "\(\mathrm{15 ÷ 5 = 10}\)" and write \(\mathrm{5(x + 10)}\).

This leads them to select Choice B. \(\mathrm{5(x + 10)}\).

Second Most Common Error:

Incomplete factoring understanding: Students partially understand factoring but don't apply it to both terms correctly.

They might factor 5 from only the first term \(\mathrm{(5x)}\) but leave 15 unchanged, getting \(\mathrm{5(x + 15)}\), or they add 5 to the constant instead of dividing by it.

This may lead them to select Choice C. \(\mathrm{5(x + 15)}\) or Choice D. \(\mathrm{5(x + 20)}\).

The Bottom Line:

This problem tests fundamental factoring skills. Success requires both recognizing when to factor and executing the arithmetic correctly when dividing each term by the common factor.