To determine which pairs of polynomials are correctly matched with their additive inverses, we need to verify whether each pair consists of a polynomial and its additive inverse. The additive inverse of a polynomial [tex]\( P(x) \)[/tex] is a polynomial [tex]\( Q(x) \)[/tex] such that [tex]\( P(x) + Q(x) = 0 \)[/tex].
Given pairs of polynomials and their potential additive inverses:
1. [tex]\( x^2 + 3x - 2 ; -x^2 - 3x + 2 \)[/tex]
2. [tex]\(-y^7 - 10 ; -y^7 + 10 \)[/tex]
3. [tex]\( 6z^5 + 6z^5 - 6z^4 ; \left(-6z^5\right) + \left(-6z^5\right) + 6z^4 \)[/tex]
4. [tex]\( x - 1 ; 1 - x \)[/tex]
5. [tex]\(\left(-5x^2\right) + (-2x) + (-10) ; 5x^2 - 2x + 10\)[/tex]
Let's check each pair:
1. [tex]\( x^2 + 3x - 2 \)[/tex] and [tex]\(-x^2 - 3x + 2 \)[/tex]
The sum is: [tex]\((x^2 + 3x - 2) + (-x^2 - 3x + 2) = x^2 - x^2 + 3x - 3x - 2 + 2 = 0\)[/tex]
Confirmed, the first pair is correct.
2. [tex]\(-y^7 - 10 \)[/tex] and [tex]\(-y^7 + 10 \)[/tex]
The sum is: [tex]\((-y^7 - 10) + (-y^7 + 10) = -y^7 - y^7 - 10 + 10 = -2y^7\)[/tex]
This is not zero, so the second pair is incorrect.
3. [tex]\( 6z^5 + 6z^5 - 6z^4 \)[/tex] and [tex]\(\left(-6z^5\right) + \left(-6z^5\right) + 6z^4 \)[/tex]
The sum is: [tex]\((6z^5 + 6z^5 - 6z^4) + \left(-6z^5 + -6z^5 + 6z^4\right) = 12z^5 - 6z^4 - 12z^5 + 6z^4 = 0\)[/tex]
Confirmed, the third pair is correct.
4. [tex]\( x - 1 \)[/tex] and [tex]\( 1 - x \)[/tex]
The sum is: [tex]\((x - 1) + (1 - x) = x - x - 1 + 1 = 0\)[/tex]
Confirmed, the fourth pair is correct.
5. [tex]\(\left(-5x^2\right) + (-2x) + (-10) \)[/tex] and [tex]\( 5x^2 - 2x + 10 \)[/tex]
The sum is: [tex]\(\left(-5x^2 + (-2x) + (-10)\right) + (5x^2 - 2x + 10) = -5x^2 + 5x^2 - 2x + (-2x) - 10 + 10 = -4x \neq 0\)[/tex]
This is not zero, so the fifth pair is incorrect.
Therefore, the polynomials that are correctly matched with their additive inverses are:
1. [tex]\( x^2 + 3x - 2 ; -x^2 - 3x + 2 \)[/tex]
2. [tex]\( x - 1 ; 1 - x \)[/tex]
3. [tex]\( 6z^5 + 6z^5 - 6z^4 ; \left(-6z^5\right) + \left(-6z^5\right) + 6z^4 \)[/tex]
