Question

match each polynomial expression to its additive inverse.
$6x^2 + x - 2$
$6x^2 - x + 2$
$-6x^2 - x - 2$
$-6x^2 + x - 2$
$6x^2 + x + 2$
$-6x^2 + x - 2$
$6x^2 - x + 2$
$-6x^2 - x + 2$

Explanation:

The additive inverse of a polynomial \( P(x) \) is \( -P(x) \), which means we change the sign of each term in the polynomial.

Step 1: Find additive inverse of \( 6x^2 + x - 2 \)

To find the additive inverse, we multiply each term by -1:
\( - (6x^2 + x - 2) = -6x^2 - x + 2 \)

Step 2: Find additive inverse of \( 6x^2 - x + 2 \)

Multiply each term by -1:
\( - (6x^2 - x + 2) = -6x^2 + x - 2 \)

Step 3: Find additive inverse of \( -6x^2 - x - 2 \)

Multiply each term by -1:
\( - (-6x^2 - x - 2) = 6x^2 + x + 2 \) Wait, no, wait. Wait, the additive inverse of \( A \) is \( -A \), so if \( A = -6x^2 - x - 2 \), then \( -A = -(-6x^2 - x - 2) = 6x^2 + x + 2 \)? Wait, no, let's correct. Wait, the additive inverse of a polynomial \( P \) is \( -P \), so for \( P = -6x^2 - x - 2 \), \( -P = -(-6x^2 - x - 2) = 6x^2 + x + 2 \)? Wait, no, that's not right. Wait, no, the additive inverse of \( P \) is the polynomial \( Q \) such that \( P + Q = 0 \). So \( Q = -P \). So for \( P = 6x^2 + x - 2 \), \( Q = -6x^2 - x + 2 \). For \( P = 6x^2 - x + 2 \), \( Q = -6x^2 + x - 2 \). For \( P = -6x^2 - x - 2 \), \( Q = 6x^2 + x + 2 \)? Wait, no, \( (-6x^2 - x - 2) + (6x^2 + x + 2) = 0 \), yes. For \( P = -6x^2 + x - 2 \), \( Q = 6x^2 - x + 2 \), because \( (-6x^2 + x - 2) + (6x^2 - x + 2) = 0 \).

Wait, let's redo:

1. Polynomial: \( 6x^2 + x - 2 \)

Additive inverse: \( - (6x^2 + x - 2) = -6x^2 - x + 2 \)

1. Polynomial: \( 6x^2 - x + 2 \)

Additive inverse: \( - (6x^2 - x + 2) = -6x^2 + x - 2 \)

1. Polynomial: \( -6x^2 - x - 2 \)

Additive inverse: \( - (-6x^2 - x - 2) = 6x^2 + x + 2 \)

1. Polynomial: \( -6x^2 + x - 2 \)

Additive inverse: \( - (-6x^2 + x - 2) = 6x^2 - x + 2 \)

Now let's match:

• \( 6x^2 + x - 2 \) matches with \( -6x^2 - x + 2 \)
• \( 6x^2 - x + 2 \) matches with \( -6x^2 + x - 2 \)
• \( -6x^2 - x - 2 \) matches with \( 6x^2 + x + 2 \)
• \( -6x^2 + x - 2 \) matches with \( 6x^2 - x + 2 \)

So the correct matches are:

• \( 6x^2 + x - 2 \) → \( -6x^2 - x + 2 \)
• \( 6x^2 - x + 2 \) → \( -6x^2 + x - 2 \)
• \( -6x^2 - x - 2 \) → \( 6x^2 + x + 2 \)
• \( -6x^2 + x - 2 \) → \( 6x^2 - x + 2 \)

Answer:

• \( 6x^2 + x - 2 \) matches with \( -6x^2 - x + 2 \)
• \( 6x^2 - x + 2 \) matches with \( -6x^2 + x - 2 \)
• \( -6x^2 - x - 2 \) matches with \( 6x^2 + x + 2 \)
• \( -6x^2 + x - 2 \) matches with \( 6x^2 - x + 2 \)