To find the additive inverse of a polynomial, we need to change the sign of every term in the polynomial. The additive inverse of a polynomial [tex]\( P(x, y) \)[/tex] is another polynomial [tex]\( -P(x, y) \)[/tex] such that when you add [tex]\( P(x, y) \)[/tex] and [tex]\( -P(x, y) \)[/tex] together, the result is zero.
Given the polynomial:
[tex]\[
-7 y^2 + x^2 y - 3 x y - 7 x^2
\][/tex]
we can find the additive inverse by changing the sign of each term. Let's go through the polynomial term by term:
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 these together, the additive inverse of the given polynomial is:
[tex]\[
7 y^2 - x^2 y + 3 x y + 7 x^2
\][/tex]
Therefore, the correct answer is:
[tex]\[
7 y^2 - x^2 y + 3 x y + 7 x^2
\][/tex]
