- Remove the parentheses and combine the like terms: \((-mn^2 + 2mn^2) + (2n^2 - 2n^2) + (-6m^2n + 3m^2n)\).
- Perform the arithmetic on the coefficients: \(1mn^2 + 0n^2 - 3m^2n\).
- The simplified expression is \(mn^2 - 3m^2n\), which corresponds to option A.
- Final Answer: \(\boxed{A}\)
Explanation
- Analysis of the Algebraic Expression
The objective is to simplify the algebraic expression \((-mn^2 + 2n^2 - 6m^2n) + (2mn^2 - 2n^2 + 3m^2n)\) by combining like terms. Like terms are those that share the exact same variables raised to the same exponents. Identifying and grouping these terms allows us to perform arithmetic operations on their coefficients to reach a simplified form.
- Grouping Like Terms
To simplify the sum, we first remove the parentheses. Since the two expressions are added together, we can simply list all the terms sequentially:
\[-mn^2 + 2n^2 - 6m^2n + 2mn^2 - 2n^2 + 3m^2n\]
Next, we group the terms that have identical variable parts:
- \(mn^2\) terms: \(-mn^2 + 2mn^2\)
- \(n^2\) terms: \(2n^2 - 2n^2\)
- \(m^2n\) terms: \(-6m^2n + 3m^2n\)
- Combining Terms and Final Simplification
Now, we perform the arithmetic for each group:
- For \(mn^2\): \(-1 + 2 = 1\), so \(-mn^2 + 2mn^2 = mn^2\).
- For \(n^2\): \(2 - 2 = 0\), so \(2n^2 - 2n^2 = 0\).
- For \(m^2n\): \(-6 + 3 = -3\), so \(-6m^2n + 3m^2n = -3m^2n\).
Combining these results, we get:
\[mn^2 + 0 - 3m^2n = mn^2 - 3m^2n\]
Comparing this result to the given options, we find it matches option A.
Examples
This simplification process is fundamental in algebra, much like sorting items in a warehouse. If you are managing inventory with different categories—say, boxes of apples (\(m^2n\)), oranges (\(mn^2\)), and pears (\(n^2\))—combining expressions like this helps you determine your total count for each type of fruit. Just as we combine like terms to simplify an equation, businesses group similar data points to manage resources effectively.