Sure, let's determine the additive inverse of the given polynomial step-by-step.
Given polynomial:
[tex]\[
-7 y^2 + x^2 y - 3 x y - 7 x^2
\][/tex]
Step 1: Understand the Additive Inverse
The additive inverse of a polynomial [tex]\(P(x, y)\)[/tex] is another polynomial that, when added to [tex]\(P(x, y)\)[/tex], results in the zero polynomial. Essentially, you are finding the negative of the given polynomial.
Step 2: Find the Additive Inverse
To find the additive inverse, we need to change the sign of each term in the polynomial.
1. The term [tex]\(-7 y^2\)[/tex] becomes [tex]\(7 y^2\)[/tex].
2. The term [tex]\(x^2 y\)[/tex] becomes [tex]\(-x^2 y\)[/tex].
3. The term [tex]\(-3 x y\)[/tex] becomes [tex]\(3 x y\)[/tex].
4. The term [tex]\(-7 x^2\)[/tex] becomes [tex]\(7 x^2\)[/tex].
Putting all the terms together, we get:
[tex]\[
7 y^2 - x^2 y + 3 x y + 7 x^2
\][/tex]
Therefore, the additive inverse of the polynomial
[tex]\[
-7 y^2 + x^2 y - 3 x y - 7 x^2
\][/tex]
is:
[tex]\[
7 y^2 - x^2 y + 3 x y + 7 x^2
\][/tex]
So, the correct answer from the options provided is:
[tex]\[
7 y^2 - x^2 y + 3 x y + 7 x^2
\][/tex]
