The directions to a problem say "write two equivalent expressions for the opposite, or additive inverse, of each polynomial).

What is an "additive inverse"?

An example problem is:

5x³ - 7x² + 3x -6

The additive inverse of x is -x, i.e. it is that number you have to add to x to get zero.

The additive inverse, also known as the opposite, of a polynomial is a polynomial that when added to the original polynomial, results in a sum of zero. In other words, it is the polynomial that cancels out the original polynomial when added together.

For the given example problem, 5x³ - 7x² + 3x -6, the additive inverse can be found by changing the sign of each term in the polynomial.

First, change the sign of every term:

-(5x³) + (-(-7x²)) + (-3x) + (-(-6))

This simplifies to:

-5x³ + 7x² - 3x + 6

Thus, the additive inverse of 5x³ - 7x² + 3x - 6 is -5x³ + 7x² - 3x + 6.

An "additive inverse" refers to a number or an expression that, when added to another number or expression, yields a sum of zero. In other words, it is the number that, when combined with a given number, results in cancellation or nullification.

To find the additive inverse of a polynomial, you need to change the signs of all the terms in the polynomial.

Let's take the example problem you provided:
5x³ - 7x² + 3x - 6

To find the additive inverse of this polynomial, you'll flip the signs of each term.

The first term, 5x³, becomes -5x³.
The second term, -7x², becomes +7x².
The third term, 3x, becomes -3x.
The fourth term, -6, becomes +6.

Thus, the additive inverse of the polynomial 5x³ - 7x² + 3x - 6 is -5x³ + 7x² - 3x + 6.

Remember, to find the additive inverse, you simply flip the signs of each term.