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Which expression is equivalent to the given polynomial expression?

$\left(9 m n-19 m^4 n\right)-\left(8 m^2+12 m^4 n+9 m n\right)$


A$-7 m^4 n+8 m^2$
B$-31 m^4 n-8 m^2$
C$-31 m^4 n+18 m n-8 m^2$
D$-7 m^4 n+18 m n-8 m^2$

Verified Answer :

  • Distribute the negative sign through the second polynomial expression: \(-(8m^2 + 12m^4n + 9mn) = -8m^2 - 12m^4n - 9mn\).
  • Combine the expanded expression with the first part: \(9mn - 19m^4n - 8m^2 - 12m^4n - 9mn\).
  • Identify and group like terms: \((9mn - 9mn) + (-19m^4n - 12m^4n) - 8m^2\).
  • Simplify the coefficients to obtain the final result: \(\boxed{-31m^4n - 8m^2}\).

Explanation

  1. Analyzing and Simplifying the Expression

The goal is to simplify the given polynomial expression:
\[(9mn - 19m^4n) - (8m^2 + 12m^4n + 9mn)\]
First, identify all the terms involved: \(9mn\), \(-19m^4n\), \(8m^2\), \(12m^4n\), and \(9mn\). To simplify, we must remove the parentheses and combine the like terms. The negative sign in front of the second set of parentheses must be distributed to each term inside.

  1. Distributing and Grouping Like Terms

Distribute the negative sign to each term inside the second parenthesis:
\[-(8m^2 + 12m^4n + 9mn) = -8m^2 - 12m^4n - 9mn\]
Now, rewrite the entire expression by combining this with the first part:
\[9mn - 19m^4n - 8m^2 - 12m^4n - 9mn\]
Next, group the like terms together to make them easier to combine:
\[(9mn - 9mn) + (-19m^4n - 12m^4n) - 8m^2\]

  1. Combining Coefficients and Final Selection

Combine the coefficients for each group of variables:

  • For the \(mn\) terms: \(9 - 9 = 0\), so the \(mn\) terms cancel out.
  • For the \(m^4n\) terms: \(-19 - 12 = -31\), so these combine to \(-31m^4n\).
  • The remaining term is \(-8m^2\).

Combining these results gives the simplified expression:
\[-31m^4n - 8m^2\]
Comparing this result with the given choices:

  • \(-7m^4n + 8m^2\)
  • \(-31m^4n - 8m^2\)
  • \(-31m^4n + 18mn - 8m^2\)
  • \(-7m^4n + 18mn - 8m^2\)

Our calculated expression matches the second option. Therefore, the equivalent expression is \(-31m^4n - 8m^2\).

Examples

In fields such as engineering or economics, simplifying complex expressions is essential for modeling systems effectively. For example, if you are modeling the total cost of two different manufacturing processes where \(m\) represents the quantity of materials and \(n\) represents labor hours, you might end up with an expression like this one. Simplifying it allows for easier calculation, budgeting, and performance analysis, ensuring that decision-makers have a clear and accurate understanding of the underlying variables.

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