To determine which pairs of polynomials are additive inverses, we add each pair and check if the result is zero. The polynomials are as follows:
1. [tex]\( x^2 + 3x - 2 \)[/tex] and [tex]\( -x^2 - 3x + 2 \)[/tex]
2. [tex]\( -y^7 - 10 \)[/tex] and [tex]\( -y^7 + 10 \)[/tex]
3. [tex]\( 6z^5 + 6z^5 - 6z^4 \)[/tex] and [tex]\( (-6z^5) + (-6z^5) + 6z^4 \)[/tex]
4. [tex]\( x - 1 \)[/tex] and [tex]\( 1 - x \)[/tex]
5. [tex]\( -5x^2 - 2x - 10 \)[/tex] and [tex]\( 5x^2 - 2x + 10 \)[/tex]
### Pair 1: [tex]\( x^2 + 3x - 2 \)[/tex] and [tex]\( -x^2 - 3x + 2 \)[/tex]
[tex]\[ (x^2 + 3x - 2) + (-x^2 - 3x + 2) = x^2 - x^2 + 3x - 3x - 2 + 2 = 0 \][/tex]
This pair is an additive inverse.
### Pair 2: [tex]\( -y^7 - 10 \)[/tex] and [tex]\( -y^7 + 10 \)[/tex]
[tex]\[ (-y^7 - 10) + (-y^7 + 10) = -y^7 - y^7 - 10 + 10 = -2y^7 \neq 0 \][/tex]
This pair is not an additive inverse.
### Pair 3: [tex]\( 6z^5 + 6z^5 - 6z^4 \)[/tex] and [tex]\( (-6z^5) + (-6z^5) + 6z^4 \)[/tex]
[tex]\[ (6z^5 + 6z^5 - 6z^4) + (-6z^5 - 6z^5 + 6z^4) = 6z^5 - 6z^5 + 6z^5 - 6z^5 - 6z^4 + 6z^4 = 0 \][/tex]
This pair is an additive inverse.
### Pair 4: [tex]\( x - 1 \)[/tex] and [tex]\( 1 - x \)[/tex]
[tex]\[ (x - 1) + (1 - x) = x - x - 1 + 1 = 0 \][/tex]
This pair is an additive inverse.
### Pair 5: [tex]\( -5x^2 - 2x - 10 \)[/tex] and [tex]\( 5x^2 - 2x + 10 \)[/tex]
[tex]\[ (-5x^2 - 2x - 10) + (5x^2 - 2x + 10) = -5x^2 + 5x^2 - 2x - 2x - 10 + 10 = -4x \neq 0 \][/tex]
This pair is not an additive inverse.
### Conclusion
The polynomials that are listed with their correct additive inverse are:
1. [tex]\( x^2 + 3x - 2 \)[/tex] and [tex]\( -x^2 - 3x + 2 \)[/tex]
3. [tex]\( 6z^5 + 6z^5 - 6z^4 \)[/tex] and [tex]\( (-6z^5) + (-6z^5) + 6z^4 \)[/tex]
4. [tex]\( x - 1 \)[/tex] and [tex]\( 1 - x \)[/tex]
