Opposite Polynomials

The opposite of a polynomial $ P(x) $ is the polynomial $ -P(x) $, obtained by reversing the sign of each term in $ P(x) $.

For example, consider the polynomial:

$$ P(x):\ 2x^2 - 3x + 2 $$

The opposite polynomial is formed by changing the sign of every term:

$$ -P(x):\ -(2x^2 - 3x + 2) $$

$$ -P(x):\ -2x^2 + 3x - 2 $$

Therefore, $ -P(x) $ is the opposite of $ P(x) $, and vice versa.

Note: The sum of a polynomial and its opposite always results in the zero polynomial. For example: $$ P(x) + (-P(x)) = (2x^2 - 3x + 2) + (-2x^2 + 3x - 2) $$ $$ = 2x^2 - 3x + 2 - 2x^2 + 3x - 2 $$ $$ = x^2(2 - 2) + x(-3 + 3) + (2 - 2) $$ $$ = 0 $$

And so on.