To determine the additive inverse of a polynomial, you need to change the sign of each term in the polynomial. Let's look at the given polynomial and its terms:
Given polynomial:
[tex]\[ -7y^2 + x^2y - 3xy - 7x^2 \][/tex]
Now, let's change the sign of each term:
1. Changing the sign of [tex]\(-7y^2\)[/tex] gives [tex]\(+7y^2\)[/tex].
2. Changing the sign of [tex]\(+x^2y\)[/tex] gives [tex]\(-x^2y\)[/tex].
3. Changing the sign of [tex]\(-3xy\)[/tex] gives [tex]\(+3xy\)[/tex].
4. Changing the sign of [tex]\(-7x^2\)[/tex] gives [tex]\(+7x^2\)[/tex].
So, the additive inverse of the polynomial [tex]\(-7y^2 + x^2y - 3xy - 7x^2\)[/tex] is:
[tex]\[ 7y^2 - x^2y + 3xy + 7x^2 \][/tex]
Now let's match this result with the provided options:
1. [tex]\[ 7y^2 - x^2 y + 3xy + 7x^2 \][/tex]
2. [tex]\[ 7y^2 + x^2 y + 3xy + 7x^2 \][/tex]
3. [tex]\[ -7y^2 - x^2 y - 3xy - 7x^2 \][/tex]
4. [tex]\[ 7y^2 + x^2 y - 3xy - 7x^2 \][/tex]
Option 1 exactly matches the additive inverse we calculated:
[tex]\[ 7y^2 - x^2 y + 3xy + 7x^2 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
