- Identify the polynomial being subtracted: \(P = -3a^2 + 2b^2 + 8ab\).
- Recall that the additive inverse \(Q\) satisfies \(P + Q = 0\), which means \(Q = -P\).
- Distribute the negative sign to negate each term: \(Q = 3a^2 - 2b^2 - 8ab\).
- Select the corresponding option: \(\boxed{c}\).
Explanation
- Identifying the Polynomial and the Goal
The problem asks for the additive inverse of the polynomial being subtracted from the expression \((3a^2 - 5ab + b^2) - (-3a^2 + 2b^2 + 8ab)\). The polynomial being subtracted is \(P = -3a^2 + 2b^2 + 8ab\). By definition, the additive inverse of a polynomial \(P\) is another polynomial \(Q\) such that \(P + Q = 0\). This implies that \(Q = -P\).
- Calculating the Additive Inverse
To find the additive inverse \(Q = -P\), we need to negate every term within the polynomial \(P = -3a^2 + 2b^2 + 8ab\).
Distributing the negative sign across the polynomial, we calculate:
\[Q = -(-3a^2 + 2b^2 + 8ab)\]
\[Q = -(-3a^2) - (2b^2) - (8ab)\]
\[Q = 3a^2 - 2b^2 - 8ab\]
Thus, the additive inverse is \(3a^2 - 2b^2 - 8ab\).
- Selecting the Correct Option
Comparing our result \(3a^2 - 2b^2 - 8ab\) with the provided options:
A\(-3a^2 - 2b^2 - 8ab\)
B\(-3a^2 + 2b^2 - 8ab\)
C\(3a^2 - 2b^2 - 8ab\)
D\(3a^2 + 2b^2 + 8ab\)
The calculated additive inverse matches option c.
Examples
The concept of additive inverses is fundamental in algebra, much like how finding the "opposite" of a number helps in balancing accounts. For example, if you have a debt of \(\$150\), which can be represented as \(-150\), the additive inverse is \(\$150\), which represents earning that amount to reach a net balance of zero. Similarly, in engineering or physics, if you define a specific force as a vector in one direction, the additive inverse represents the equal and opposite force that maintains equilibrium in the system.