What is the additive inverse of polynomial?

In an additive group

, the additive inverse of an element

is the element

such that

, where 0 is the additive identity of

. Usually, the additive inverse of

is denoted

, as in the additive group of integers

, of rationals

, of real numbers

, and of complex numbers

, where

The same notation with the minus sign is used to denote the additive inverse of a vector,

of a polynomial,

of a matrix

and, in general, of any element in an abstract vector space or a module.

This is very easy when using the augmented-matrix form of the extended Euclidean algorithm, i.e. we perform the Euclidean algorthm while keeping track of each remainders expression as a linear combination of $f$ and $g$ as follows.

$\begin{eqnarray} (1)&& &&f = x^3\!+2x+1 &\!\!=&\, \left<\,\color{#c00}1,\,\color{#0a0}0\,\right>\quad\ \ \, {\rm i.e.}\ \qquad f\, =\ \color{#c00}1\cdot f\, +\, \color{#0a0}0\cdot g\\ (2)&& &&\qquad\ \, g =x^2\!+1 &\!\!=&\, \left<\,\color{#c00}0,\,\color{#0a0}1\,\right>\quad\ \ \,{\rm i.e.}\ \qquad g\, =\ \color{#c00}0\cdot f\, +\, \color{#0a0}1\cdot g\\ (3)&=&(1)-x(2)\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! &&\qquad\qquad\ \ x+1 \,&\!\!=&\, \left<\,\color{#c00}1,\,\color{#0a0}{-x}\,\right>\ \ \ {\rm i.e.}\quad x\!+\!1\, =\, \color{#c00}1\cdot f\,\color{#0c0}{-\,x}\cdot g\\ (4)&=&(2)+(1\!-\!x)(3)\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! &&\qquad\qquad\qquad\ 2 \,&\!\!=&\, \left<\,\color{#c00}{1\!-\!x},\,\color{#0a0}{1\!-\!x+x^2}\,\right>\\ \end{eqnarray}$

Hence the prior line implies: $\,\ 2\, =\, (\color{#c00}{1\!-\!x})f + (\color{#0a0}{1\!-\!x\!+\!x^2})g,\, $ so reducing this mod $f$ and $3$

we get in $\,\Bbb Z_3[x] \bmod f\!:\,\ {-}1\equiv 2 \equiv (\color{#0a0}{1\!-\!x\!+\!x^2})g\ \Rightarrow\ \bbox[6px,border:1px solid red]{g^{-1}\equiv\, {-}(\color{#0a0}{1\!-\!x\!+\!x^2})}$

Remark $\ $ Generally, this method is easier to memorize and much less error-prone than the alternative "back-substitution" method.

This is a special-case of Hermite/Smith row/column reduction of matrices to triangular/diagonal normal form, using the division/Euclidean algorithm to reduce entries modulo pivots. Though one can understand this knowing only the analogous linear algebra elimination techniques, it will become clearer when one studies modules - which, informally, generalize vector spaces by allowing coefficients from rings vs. fields. In particular, these results are studied when one studies normal forms for finitely-generated modules over a PID, e.g. when one studies linear systems of equations with coefficients in the non-field! polynomial ring $\rm F[x],$ for $\rm F$ a field, as above.