Which expression is equivalent to the expression 4xy^2 - 2x^2y + 7xy^2?

1. TRANSLATE the problem information

• Given expression: \(4xy^2 - 2x^2y + 7xy^2\)
• Task: Find an equivalent expression by combining like terms

2. INFER which terms can be combined

• Like terms must have identical variable parts (same variables with same exponents)
• Compare each term's variable part:
  • First term: \(4xy^2\) (variable part is \(xy^2\))
  • Second term: \(-2x^2y\) (variable part is \(x^2y\))
  • Third term: \(7xy^2\) (variable part is \(xy^2\))
• The first and third terms (\(4xy^2\) and \(7xy^2\)) are like terms
• The second term (\(-2x^2y\)) is different and cannot be combined

3. SIMPLIFY by combining like terms

• Combine the coefficients of like terms: \(4 + 7 = 11\)
• Keep the variable part unchanged: \(xy^2\)
• Combined like terms: \(11xy^2\)
• Keep the unlike term as is: \(-2x^2y\)

4. Write the final equivalent expression

• \(11xy^2 - 2x^2y\)

Answer: D. \(11xy^2 - 2x^2y\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't carefully distinguish between \(x^2y\) and \(xy^2\), thinking they're the same because they contain the same letters.

They might reason: "All three terms have x and y, so I can combine all the coefficients: \(4 + (-2) + 7 = 9\)" and then guess at the variable part, leading them to select Choice B (\(9xy^2\)).

Second Most Common Error:

Incomplete SIMPLIFY execution: Students correctly identify and combine the like terms (\(4xy^2 + 7xy^2 = 11xy^2\)) but forget to include the unlike term (\(-2x^2y\)) in their final answer.

This leads them to select Choice C (\(11xy^2\)), thinking they're done after combining just the like terms.

The Bottom Line:

Success requires careful attention to the exact variable parts of each term. The difference between \(x^2y\) and \(xy^2\) is crucial - even though they contain the same variables, the different exponents make them completely different terms that cannot be combined.