Let's verify which pairs of polynomials are correct additive inverses of each other. When adding a polynomial to its additive inverse, the result should be zero.
1. For the pair [tex]\( x^2 + 3x - 2 \; \text{and} \; -x^2 - 3x + 2 \)[/tex]:
[tex]\[
(x^2 + 3x - 2) + (-x^2 - 3x + 2) = x^2 + 3x - 2 - x^2 - 3x + 2 = 0
\][/tex]
The result is zero, indicating that they are correct additive inverses.
2. For the pair [tex]\( -y^7 - 10 \; \text{and} \; -y^7 + 10 \)[/tex]:
[tex]\[
(-y^7 - 10) + (-y^7 + 10) = -y^7 - 10 - y^7 + 10 = -2y^7 \neq 0
\][/tex]
The result is [tex]\(-2y^7\)[/tex], indicating that they are not correct additive inverses.
3. For the pair [tex]\( 6z^5 + 6z^5 - 6z^4 \; \text{and} \; (-6z^5) + (-6z^5) + 6z^4 \)[/tex]:
[tex]\[
(6z^5 + 6z^5 - 6z^4) + (-6z^5 + (-6z^5) + 6z^4) = 6z^5 + 6z^5 - 6z^4 - 6z^5 - 6z^5 + 6z^4 = 0
\][/tex]
The result is zero, indicating that they are correct additive inverses.
4. For the pair [tex]\( x - 1 \; \text{and} \; 1 - x \)[/tex]:
[tex]\[
(x - 1) + (1 - x) = x - 1 + 1 - x = 0
\][/tex]
The result is zero, indicating that they are correct additive inverses.
5. For the pair [tex]\( -5x^2 + (-2x) + (-10) \; \text{and} \; 5x^2 - 2x + 10 \)[/tex]:
[tex]\[
(-5x^2 + (-2x) + (-10)) + (5x^2 - 2x + 10) = -5x^2 - 2x - 10 + 5x^2 - 2x + 10 = -4x \neq 0
\][/tex]
The result is [tex]\(-4x\)[/tex], indicating that they are not correct additive inverses.
Conclusion:
The polynomials that are correctly listed with their additive inverses are:
[tex]\[ x^2 + 3x - 2 \; \text{and} \; -x^2 - 3x + 2 \][/tex]
[tex]\[ 6z^5 + 6z^5 - 6z^4 \; \text{and} \; (-6z^5) + (-6z^5) + 6z^4 \][/tex]
[tex]\[ x - 1 \; \text{and} \; 1 - x \][/tex]
Thus, the correct pairs are 1, 3, and 4.
