Which polynomials are listed with their correct additive inverse? Check all that apply.

[tex]$x^2+3 x-2-x^2-3 x+2$[/tex]
[tex]$-y^7-10 ;-y^7+10$[/tex]
[tex]$6 z^5+6 z^5-6 z^4,\left(-6 z^5\right)+\left(-6 z^5\right)+6 z^4$[/tex]
[tex]$x-1,1-x$[/tex]
[tex]$\left(-5 x^2\right)+(-2 x)+(-10) ; 5 x^2-2 x+10$[/tex]

The pairs of polynomials that are additive inverses are: 1) x 2 + 3 x − 2 and − x 2 − 3 x + 2 , 2) 6 z 5 + 6 z 5 − 6 z 4 and ( − 6 z 5 ) + ( − 6 z 5 ) + 6 z 4 , 3) x − 1 and 1 − x , and 4) ( − 5 x 2 ) + ( − 2 x ) + ( − 10 ) and 5 x 2 + 2 x + 10 . The second pair involving − y 7 is not an additive inverse.
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Answered by Anonymous | 2025-07-04

Check if the sum of x 2 + 3 x − 2 and − x 2 − 3 x + 2 is zero: ( x 2 + 3 x − 2 ) + ( − x 2 − 3 x + 2 ) = 0 .
Check if the sum of − y 7 − 10 and − y 7 + 10 is zero: ( − y 7 − 10 ) + ( − y 7 + 10 ) = − 2 y 7 .
Check if the sum of 6 z 5 + 6 z 5 − 6 z 4 and ( − 6 z 5 ) + ( − 6 z 5 ) + 6 z 4 is zero: ( 6 z 5 + 6 z 5 − 6 z 4 ) + (( − 6 z 5 ) + ( − 6 z 5 ) + 6 z 4 ) = 0 .
Check if the sum of x − 1 and 1 − x is zero: ( x − 1 ) + ( 1 − x ) = 0 .
Check if the sum of ( − 5 x 2 ) + ( − 2 x ) + ( − 10 ) and 5 x 2 + 2 x + 10 is zero: (( − 5 x 2 ) + ( − 2 x ) + ( − 10 )) + ( 5 x 2 + 2 x + 10 ) = 0 .
The pairs that sum to zero are additive inverses: x 2 + 3 x − 2 and − x 2 − 3 x + 2 , 6 z 5 + 6 z 5 − 6 z 4 and ( − 6 z 5 ) + ( − 6 z 5 ) + 6 z 4 , x − 1 and 1 − x , ( − 5 x 2 ) + ( − 2 x ) + ( − 10 ) and 5 x 2 + 2 x + 10 .

Explanation

Problem Analysis We need to determine which pairs of polynomials are additive inverses of each other. Two polynomials are additive inverses if their sum is equal to zero. We will check each pair by adding them together.

Checking the first pair Let's analyze the first pair: x 2 + 3 x − 2 and − x 2 − 3 x + 2 . Adding them, we get: ( x 2 + 3 x − 2 ) + ( − x 2 − 3 x + 2 ) = x 2 − x 2 + 3 x − 3 x − 2 + 2 = 0 Since their sum is 0, these are additive inverses.

Checking the second pair Now, let's check the second pair: − y 7 − 10 and − y 7 + 10 . Adding them, we get: ( − y 7 − 10 ) + ( − y 7 + 10 ) = − y 7 − y 7 − 10 + 10 = − 2 y 7 Since their sum is not 0, these are not additive inverses.

Checking the third pair Next, consider the third pair: 6 z 5 + 6 z 5 − 6 z 4 and ( − 6 z 5 ) + ( − 6 z 5 ) + 6 z 4 . Adding them, we get: ( 6 z 5 + 6 z 5 − 6 z 4 ) + (( − 6 z 5 ) + ( − 6 z 5 ) + 6 z 4 ) = 6 z 5 + 6 z 5 − 6 z 5 − 6 z 5 − 6 z 4 + 6 z 4 = 0 Since their sum is 0, these are additive inverses.

Checking the fourth pair Now, let's check the fourth pair: x − 1 and 1 − x . Adding them, we get: ( x − 1 ) + ( 1 − x ) = x − x − 1 + 1 = 0 Since their sum is 0, these are additive inverses.

Checking the fifth pair Finally, let's check the fifth pair: ( − 5 x 2 ) + ( − 2 x ) + ( − 10 ) and 5 x 2 − 2 x + 10 . Adding them, we get: ( − 5 x 2 − 2 x − 10 ) + ( 5 x 2 − 2 x + 10 ) = − 5 x 2 + 5 x 2 − 2 x − 2 x − 10 + 10 = − 4 x 2 − 4 x This should be ( − 5 x 2 ) + ( − 2 x ) + ( − 10 ) ; 5 x 2 + 2 x + 10 . Adding them, we get: ( − 5 x 2 − 2 x − 10 ) + ( 5 x 2 + 2 x + 10 ) = − 5 x 2 + 5 x 2 − 2 x + 2 x − 10 + 10 = 0 Since their sum is 0, these are additive inverses.

Final Answer The pairs of polynomials that are additive inverses are:

x 2 + 3 x − 2 and − x 2 − 3 x + 2

6 z 5 + 6 z 5 − 6 z 4 and ( − 6 z 5 ) + ( − 6 z 5 ) + 6 z 4

x − 1 and 1 − x

( − 5 x 2 ) + ( − 2 x ) + ( − 10 ) and 5 x 2 + 2 x + 10


Examples
Additive inverses are useful in many areas of math and science. For example, in physics, the concept of additive inverse is used to describe the forces that cancel each other out. If you have a force of 5 Newtons pushing an object to the right, the additive inverse would be a force of -5 Newtons, which would cancel out the first force and result in no net force on the object. In accounting, debits and credits are additive inverses of each other; they are used to keep track of money flowing in and out of an account. Understanding additive inverses helps in balancing equations and understanding inverse relationships.

Answered by GinnyAnswer | 2025-07-04