Which expression is equivalent to 15b^2 - 6b^2 + 3b^2? 6b^2 9b^2 12b^2 18b^2...

1. INFER what the problem is asking

• We need to find an equivalent expression for \(\mathrm{15b^2 - 6b^2 + 3b^2}\)
• "Equivalent" means the expressions have the same value for any value of b

2. INFER the approach

• All three terms have the same variable part: \(\mathrm{b^2}\)
• These are like terms, so we can combine them by working with their coefficients
• Keep the variable part \(\mathrm{(b^2)}\) and add/subtract the numbers in front

3. SIMPLIFY by combining the like terms

• \(\mathrm{15b^2 - 6b^2 + 3b^2 = (15 - 6 + 3)b^2}\)
• Calculate the coefficients: \(\mathrm{15 - 6 + 3 = 9 + 3 = 12}\)
• Result: \(\mathrm{12b^2}\)

Answer: (C) \(\mathrm{12b^2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making sign errors when computing \(\mathrm{15 - 6 + 3}\)

Students might mishandle the signs and compute:

• \(\mathrm{15 - 6 - 3 = 6}\), leading them to select Choice (A) \(\mathrm{(6b^2)}\)
• \(\mathrm{15 + 6 + 3 = 24}\) (not an option, causing confusion)
• \(\mathrm{15 + 6 - 3 = 18}\), leading them to select Choice (D) \(\mathrm{(18b^2)}\)

Second Most Common Error:

Incomplete SIMPLIFY process: Stopping partway through the calculation

A student might compute \(\mathrm{15 - 6 = 9}\) and forget about the "\(\mathrm{+ 3b^2}\)" term entirely, leading them to select Choice (B) \(\mathrm{(9b^2)}\).

The Bottom Line:

This problem tests careful arithmetic with signed numbers. The algebra concept is straightforward, but execution errors in basic arithmetic cause most mistakes.