To determine which polynomials have the correct additive inverse, we need to check if the sum of the polynomial and its provided inverse equals zero. Let's go through each pair step by step:
1. [tex]\[ x^2 + 3x - 2 \][/tex]
[tex]\[ -x^2 - 3x + 2 \][/tex]
To find the sum:
[tex]\[ (x^2 + 3x - 2) + (-x^2 - 3x + 2) \][/tex]
Combine like terms:
[tex]\[ x^2 - x^2 + 3x - 3x - 2 + 2 = 0 \][/tex]
So, the additive inverse is correct for this pair.
2. [tex]\[ -y^7 - 10 \][/tex]
[tex]\[ -y^7 + 10 \][/tex]
To find the sum:
[tex]\[ (-y^7 - 10) + (-y^7 + 10) \][/tex]
Combine like terms:
[tex]\[ -y^7 - y^7 - 10 + 10 = -2y^7 \][/tex]
This does not simplify to zero. Hence, the additive inverse is incorrect for this pair.
3. [tex]\[ 6z^5 + 6z^5 - 6z^4 \][/tex]
[tex]\[ -6z^5 + (-6z^5) + 6z^4 \][/tex]
To find the sum:
[tex]\[ (6z^5 + 6z^5 - 6z^4) + (-6z^5 + -6z^5 + 6z^4) \][/tex]
Combine like terms:
[tex]\[ 6z^5 - 6z^5 + 6z^5 - 6z^5 - 6z^4 + 6z^4 = 0 \][/tex]
So, the additive inverse is correct for this pair.
4. [tex]\[ x - 1 \][/tex]
[tex]\[ 1 - x \][/tex]
To find the sum:
[tex]\[ (x - 1) + (1 - x) \][/tex]
Combine like terms:
[tex]\[ x - x - 1 + 1 = 0 \][/tex]
So, the additive inverse is correct for this pair.
5. [tex]\[ -5x^2 - 2x - 10 \][/tex]
[tex]\[ 5x^2 - 2x + 10 \][/tex]
To find the sum:
[tex]\[ (-5x^2 - 2x - 10) + (5x^2 - 2x + 10) \][/tex]
Combine like terms:
[tex]\[ -5x^2 + 5x^2 - 2x - 2x - 10 + 10 = -4x \][/tex]
This does not simplify to zero. Hence, the additive inverse is incorrect for this pair.
In summary, the polynomials that are correctly paired with their additive inverse are:
1. [tex]\( x^2 + 3x - 2 \; ; \; -x^2 - 3x + 2 \)[/tex]
3. [tex]\( 6z^5 + 6z^5 - 6z^4 \; ; \; -6z^5 + -6z^5 + 6z^4 \)[/tex]
4. [tex]\( x - 1 \; ; \; 1 - x \)[/tex]
