Multiplicative inverse of $x+f(x)$ in $\Bbb Q[x]/(f(x))$ So I have $f(x) = x^3-2$ and I have to find the multiplicative inverse of $x + f(x)$ in $\mathbb{Q}[x]/(f(x))$. I'm slightly confused as to how to represent $x + (f(x))$ in $\mathbb{Q}[x]/(f(x))$. Would I be just finding the inverse of $x+1$? How would I do that? Thanks a lot!
Note $x^3=2$ in the quotient, so $x\cdot x^2=2$ (in the quotient) so $x\cdot {1\over 2}x^2=1$.
$1=(x+1)(\frac{x^2-x+1}{3})-\frac{x^3-2}{3}$, then $\frac{x^2-x+1}{3}$ is the multiplicative inverse of $x+1+\langle f\rangle$ in $\frac{\mathbb{Q}[x]}{\langle f\rangle}.$