- The additive inverse of a polynomial \(P\) is obtained by multiplying every term by \(-1\).
- Applying this to \(P = -7y^2 + x^2y - 3xy - 7x^2\) results in \(7y^2 - x^2y + 3xy + 7x^2\).
- This expression corresponds to option c.
- The final answer is \(\boxed{7y^2 - x^2y + 3xy + 7x^2}\).
Explanation
- Understanding the Additive Inverse
To find the additive inverse of a polynomial \(P(x, y)\), we seek a polynomial \(Q(x, y)\) such that their sum equals zero, specifically \(P(x, y) + Q(x, y) = 0\). This implies that \(Q(x, y) = -P(x, y)\). Given our polynomial \(P(x, y) = -7y^2 + x^2y - 3xy - 7x^2\), we need to negate each individual term to find its additive inverse.
- Calculating the Additive Inverse
We calculate \(Q(x, y) = -(-7y^2 + x^2y - 3xy - 7x^2)\). By distributing the negative sign across all terms within the parentheses, we change the sign of each coefficient:
- The term \(-7y^2\) becomes \(-(-7y^2) = 7y^2\).
- The term \(x^2y\) becomes -(x^2y) = -x^2y\).
- The term \(-3xy\) becomes \(-(-3xy) = 3xy\).
- The term \(-7x^2\) becomes \(-(-7x^2) = 7x^2\).
Combining these, we get \(Q(x, y) = 7y^2 - x^2y + 3xy + 7x^2\).
- Identifying the Correct Option
Comparing the result \(7y^2 - x^2y + 3xy + 7x^2\) with the given options:
A\(-7y^2 - x^2y - 3xy - 7x^2\)
B\(7y^2 + x^2y - 3xy - 7x^2\)
C\(7y^2 - x^2y + 3xy + 7x^2\)
D\(7y^2 + x^2y + 3xy + 7x^2\)
The result matches option c. The additive inverse is \(\boxed{7y^2 - x^2y + 3xy + 7x^2}\).
Examples
In financial management, calculating the net balance of transactions often involves finding the additive inverse. For instance, if you have a debt of \(\$100\) (represented as \(-100\)), the amount needed to neutralize the balance (bring it to \(\$0\)) is its additive inverse, which is \(\$100\). Similarly, when balancing equations in accounting or physics, finding the inverse allows you to cancel out specific terms to isolate a variable or simplify an expression, ensuring the system remains in equilibrium.