Question

which polynomials are listed with their correct additive inverse? check all that apply.
$x^2 + 3x - 2; -x^2 - 3x + 2$
$-y^7 - 10; -y^7 + 10$
$6x^5 + 6x^5 - 6x^4; (-6x^5) + (-6x^5) + 6x^4$
$x - 1; 1 - x$
$(-5x^2) + (-2x) + (-10); 5x^2 - 2x + 10$

Explanation:

Step1: Define additive inverse

The additive inverse of a polynomial $P(x)$ is $-P(x)$, such that $P(x) + (-P(x)) = 0$. We check each pair by adding them.

Step2: Check first pair

Add $x^2 + 3x - 2$ and $-x^2 - 3x + 2$:
$$(x^2 + 3x - 2) + (-x^2 - 3x + 2) = (x^2 - x^2) + (3x - 3x) + (-2 + 2) = 0$$
This is a correct pair.

Step3: Check second pair

Add $-y^7 - 10$ and $-y^7 + 10$:

$$(-y^7 - 10) + (-y^7 + 10) = -2y^7 + 0 = -2y^7 eq 0$$

This is incorrect.

Step4: Check third pair

Add $6z^5 + 6z^5 - 6z^4$ and $(-6z^5) + (-6z^5) + 6z^4$:
$$(6z^5 + 6z^5 - 6z^4) + (-6z^5 -6z^5 + 6z^4) = (12z^5 -6z^4) + (-12z^5 +6z^4) = 0$$
This is a correct pair.

Step5: Check fourth pair

Add $x - 1$ and $1 - x$:
$$(x - 1) + (1 - x) = (x - x) + (-1 + 1) = 0$$
This is a correct pair.

Step6: Check fifth pair

Add $(-5x^2) + (-2x) + (-10)$ and $5x^2 - 2x + 10$:

$$(-5x^2 -2x -10) + (5x^2 -2x +10) = (-5x^2+5x^2) + (-2x-2x) + (-10+10) = -4x eq 0$$

This is incorrect.

Answer:

• $x^2 + 3x - 2; -x^2 - 3x + 2$
• $6z^5 + 6z^5 - 6z^4; (-6z^5) + (-6z^5) + 6z^4$
• $x - 1; 1 - x$