Which expression is equivalent to 15x^2 - 25x? \(5\mathrm{x}(5 - 3\mathrm{x})\) \(5\mathrm{x}(3\mathrm{x} - 5)\) \(5\mathrm{x}(3\mathrm{x} + 5)\) \(...

1. INFER the approach needed

• Looking at \(15\mathrm{x}^2 - 25\mathrm{x}\), I can see both terms share common factors
• Strategy: Find the greatest common factor (GCF) and factor it out
• This will create an equivalent expression in factored form

2. SIMPLIFY by finding the GCF

• For coefficients: Find GCD of 15 and 25
  • Factors of 15: 1, 3, 5, 15
  • Factors of 25: 1, 5, 25
  • GCD = 5
• For variables: Find GCF of \(\mathrm{x}^2\) and \(\mathrm{x}\)
  • The lowest power of x present is \(\mathrm{x}^1 = \mathrm{x}\)
• Overall GCF = \(5\mathrm{x}\)

3. SIMPLIFY by factoring out the GCF

• Divide each term by \(5\mathrm{x}\):
  • \((15\mathrm{x}^2) ÷ (5\mathrm{x}) = 3\mathrm{x}\)
  • \((-25\mathrm{x}) ÷ (5\mathrm{x}) = -5\)
• Write in factored form: \(5\mathrm{x}(3\mathrm{x} - 5)\)

4. SIMPLIFY by verifying through distribution

• Check: \(5\mathrm{x}(3\mathrm{x} - 5) = 5\mathrm{x} \cdot 3\mathrm{x} + 5\mathrm{x} \cdot (-5) = 15\mathrm{x}^2 - 25\mathrm{x}\) ✓
• This matches our original expression

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make sign errors when factoring out the GCF, particularly with the negative term. They might incorrectly get \(5\mathrm{x}(3\mathrm{x} + 5)\) by losing track of the negative sign from \(-25\mathrm{x}\).

This leads them to select Choice C (\(5\mathrm{x}(3\mathrm{x} + 5)\)).

Second Most Common Error:

Incomplete INFER reasoning: Students recognize they need to factor but confuse the order of terms. They might reverse the expression to get \(5\mathrm{x}(5 - 3\mathrm{x})\), thinking this is equivalent but not realizing the sign change affects the final result.

This leads them to select Choice A (\(5\mathrm{x}(5 - 3\mathrm{x})\)).

The Bottom Line:

Factoring requires careful attention to both the arithmetic of finding the GCF and the algebra of maintaining correct signs throughout the process. The key insight is that factoring and distributing are inverse operations - if you factor correctly, distribution should give you back the original expression.