Simplify the following expression:
[tex]$\begin{array}{l}\left(4 m^3+6\right)-\left(-2 m^3+m^2+3\right) \\ 6 m^3+5 m^2-3 \\ 6 m^3-m^2+3 \\ 6 m^3+m^2+3 \\ 2 m^3+m^2+9\end{array}$[/tex]
Answer (2)
Distribute the negative sign: ( 4 m 3 + 6 ) − ( − 2 m 3 + m 2 + 3 ) = 4 m 3 + 6 + 2 m 3 − m 2 − 3 .
Combine the m 3 terms: 4 m 3 + 2 m 3 = 6 m 3 .
Combine the constant terms: 6 − 3 = 3 .
The simplified expression is 6 m 3 − m 2 + 3 .
Explanation
Understanding the Problem We are asked to simplify the expression ( 4 m 3 + 6 ) − ( − 2 m 3 + m 2 + 3 ) and choose the correct answer from the given options. This involves distributing the negative sign and combining like terms to arrive at the simplified expression.
Distributing the Negative Sign First, distribute the negative sign in the second term: ( 4 m 3 + 6 ) − ( − 2 m 3 + m 2 + 3 ) = 4 m 3 + 6 + 2 m 3 − m 2 − 3
Combining Like Terms Next, combine like terms. We have the m 3 terms, the m 2 term, and the constant terms: ( 4 m 3 + 2 m 3 ) − m 2 + ( 6 − 3 ) Combining the m 3 terms, 4 m 3 + 2 m 3 = 6 m 3 .
The m 2 term is just − m 2 .
Combining the constant terms, 6 − 3 = 3 .
So the simplified expression is: 6 m 3 − m 2 + 3
Choosing the Correct Answer Now, we compare our simplified expression 6 m 3 − m 2 + 3 with the given options:
6 m 3 + 5 m 2 − 3 (Incorrect)
6 m 3 − m 2 + 3 (Correct)
6 m 3 + m 2 + 3 (Incorrect)
2 m 3 + m 2 + 9 (Incorrect)
The correct answer is 6 m 3 − m 2 + 3 .
Examples
Polynomial expressions are used in various fields such as physics, engineering, and computer science. For example, in physics, polynomials can be used to model the trajectory of a projectile or the behavior of electrical circuits. In computer graphics, polynomials are used to create curves and surfaces. Simplifying polynomial expressions allows us to analyze and understand these models more easily. For instance, if we have an expression representing the cost of manufacturing a certain item, simplifying it can help us determine the most efficient way to reduce costs.
To simplify the expression ( 4 m 3 + 6 ) − ( − 2 m 3 + m 2 + 3 ) , we distribute the negative sign, combine like terms, and find that the simplified result is 6 m 3 − m 2 + 3 . The correct answer from the provided options is 6 m 3 − m 2 + 3 .
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