Answer :
To determine whether the polynomials listed are additive inverses of each other, we need to check if their sum equals zero. The additive inverse of any polynomial \( P(x) \) is \( -P(x) \). We will check each pair step-by-step.
### Pair 1: \( x^2 + 3x - 2 \) and \( -x^2 - 3x + 2 \)
[tex]\[ (x^2 + 3x - 2) + (-x^2 - 3x + 2) = x^2 - x^2 + 3x - 3x - 2 + 2 = 0 \][/tex]
These two polynomials are indeed additive inverses. Correct.
### Pair 2: \( -y^7 - 10 \) and \( -y^7 + 10 \)
[tex]\[ (-y^7 - 10) + (-y^7 + 10) = -y^7 - y^7 - 10 + 10 = -2y^7 \neq 0 \][/tex]
These two polynomials are not additive inverses. Incorrect.
### Pair 3: \( 6z^5 + 6z^5 - 6z^4 \) and \( -6z^5 - 6z^5 + 6z^4 \)
[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]
These two polynomials are additive inverses. Correct.
### Pair 4: \( x - 1 \) and \( 1 - x \)
[tex]\[ (x - 1) + (1 - x) = x - x - 1 + 1 = 0 \][/tex]
These two polynomials are additive inverses. Correct.
### Pair 5: \( -5x^2 - 2x - 10 \) and \( 5x^2 - 2x + 10 \)
[tex]\[ (-5x^2 - 2x - 10) + (5x^2 - 2x + 10) = -5x^2 + 5x^2 - 2x - 2x - 10 + 10 = 0 - 4x = -4x \neq 0 \][/tex]
These two polynomials are not additive inverses. Incorrect.
### Summary:
The polynomials that are correctly listed with their additive inverses are:
1. \( x^2 + 3x - 2 \) and \( -x^2 - 3x + 2 \)
2. \( 6z^5 + 6z^5 - 6z^4 \) and \( -6z^5 - 6z^5 + 6z^4 \)
3. \( x - 1 \) and \( 1 - x \)
Thus, the correct pairs are:
- \( x^2 + 3x - 2 \) and \( -x^2 - 3x + 2 \)
- \( 6z^5 + 6z^5 - 6z^4 \) and \( -6z^5 - 6z^5 + 6z^4 \)
- [tex]\( x - 1 \)[/tex] and [tex]\( 1 - x \)[/tex]
### Pair 1: \( x^2 + 3x - 2 \) and \( -x^2 - 3x + 2 \)
[tex]\[ (x^2 + 3x - 2) + (-x^2 - 3x + 2) = x^2 - x^2 + 3x - 3x - 2 + 2 = 0 \][/tex]
These two polynomials are indeed additive inverses. Correct.
### Pair 2: \( -y^7 - 10 \) and \( -y^7 + 10 \)
[tex]\[ (-y^7 - 10) + (-y^7 + 10) = -y^7 - y^7 - 10 + 10 = -2y^7 \neq 0 \][/tex]
These two polynomials are not additive inverses. Incorrect.
### Pair 3: \( 6z^5 + 6z^5 - 6z^4 \) and \( -6z^5 - 6z^5 + 6z^4 \)
[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]
These two polynomials are additive inverses. Correct.
### Pair 4: \( x - 1 \) and \( 1 - x \)
[tex]\[ (x - 1) + (1 - x) = x - x - 1 + 1 = 0 \][/tex]
These two polynomials are additive inverses. Correct.
### Pair 5: \( -5x^2 - 2x - 10 \) and \( 5x^2 - 2x + 10 \)
[tex]\[ (-5x^2 - 2x - 10) + (5x^2 - 2x + 10) = -5x^2 + 5x^2 - 2x - 2x - 10 + 10 = 0 - 4x = -4x \neq 0 \][/tex]
These two polynomials are not additive inverses. Incorrect.
### Summary:
The polynomials that are correctly listed with their additive inverses are:
1. \( x^2 + 3x - 2 \) and \( -x^2 - 3x + 2 \)
2. \( 6z^5 + 6z^5 - 6z^4 \) and \( -6z^5 - 6z^5 + 6z^4 \)
3. \( x - 1 \) and \( 1 - x \)
Thus, the correct pairs are:
- \( x^2 + 3x - 2 \) and \( -x^2 - 3x + 2 \)
- \( 6z^5 + 6z^5 - 6z^4 \) and \( -6z^5 - 6z^5 + 6z^4 \)
- [tex]\( x - 1 \)[/tex] and [tex]\( 1 - x \)[/tex]
