 # Find the multiplicative inverse of x^2+(x^3−x+2) in the quotient F_3[x]/(x^3−x+2) Zachariah Norris 2022-09-18 Answered
Find the multiplicative inverse of $\phantom{\rule{thinmathspace}{0ex}}{x}^{2}+\left({x}^{3}-x+2\right)$ in the quotient $\phantom{\rule{thinmathspace}{0ex}}{F}_{3}\left[x\right]/\left({x}^{3}-x+2\right)$
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You wrote you can find f,g such that ... and that's what you need to do. This is just th e(extended) Euclidean algorithm as known from integers, but requiring polynomial divisions. Thius
$\begin{array}{rl}{x}^{3}-x+2& =x\cdot {x}^{2}-\left(x-2\right)\\ {x}^{2}& =\left(x-2\right)\cdot x+2x\\ x-2& =2x\cdot \frac{1}{2}-2=2x\cdot 2+1\end{array}$
(where only the last step is aware of us working in ${\mathbb{F}}_{3}$). From this with $p\left(x\right)={x}^{3}-x+2,q\left(x\right)={x}^{2}$ we obtain step by step:
$\begin{array}{rl}x-2& =xq-p\\ 2x& =q-x\left(xq-p\right)\\ 1& =\left(xq-p\right)-2\left(q-x\left(xq-p\right)\right)\end{array}$
So
$\left(x+2\right)\cdot p\left(x\right)+\left(2{x}^{2}+x+1\right)\cdot q\left(x\right)=1$