Which expression is equivalent to \(19(\mathrm{x}^2 - 7)\)?

1. TRANSLATE the problem information

• Given: The expression \(19(x^2 - 7)\)
• Need to find: An equivalent expression from the answer choices

2. SIMPLIFY using the distributive property

• Apply the distributive property: \(a(b - c) = ab - ac\)
• Distribute 19 to both terms inside the parentheses:
  • \(19(x^2 - 7) = 19 \cdot x^2 - 19 \cdot 7\)
• Calculate the arithmetic:
  • \(19 \cdot x^2 = 19x^2\)
  • \(19 \cdot 7 = 133\)
• Final result: \(19x^2 - 133\)

Answer: A. \(19x^2 - 133\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making an arithmetic mistake when calculating \(19 \times 7\)

Students might rush through the multiplication and calculate \(19 \times 7 = 26\) instead of 133. This happens when they confuse \(19 \times 7\) with simpler calculations or make a basic computational error.

This may lead them to select Choice B (\(19x^2 - 26\))

Second Most Common Error:

Incomplete SIMPLIFY application: Not fully applying the distributive property

Some students understand they need to distribute but don't complete the process. They might think that distributing 19 only applies to the first term, leaving the second term unchanged, resulting in \(19x^2 - 7\).

This may lead them to select Choice C (\(19x^2 - 7\))

The Bottom Line:

This problem tests whether students can correctly apply the distributive property and perform accurate arithmetic. The key is methodically distributing to both terms and carefully calculating \(19 \times 7 = 133\).