Expressiong $\frac{t+2}{t^3+3}$ in the form $a_o+a_1t+...+a_4t^4$, where $t$ is a root of $x^5+2x+2$ Expressing $\frac{t+2}{t^3+3}$ in the form $a_o+a_1t+...+a_4t^4$, where $t$ is a root of $x^5+2x+2$.
So i can deal with the numerator, but how do I get rid of the denomiator to get it into the correct form? Thanks in advance!

Using the Euclidean algorithm for computing $\gcd(x^3+3,x^5+2x+2)$, we get
$$
367=(10 x^4 - 31 x^3 - 14 x^2 - 30 x + 113)(x^3+3)+(-10 x^2 + 31 x + 14)(x^5+2x+2)
$$
and so
$$
367=(10 t^4 - 31 t^3 - 14 t^2 - 30 t + 113)(t^3+3)
$$
Thus,
$$
\begin{align}
367\frac{t+2}{t^3+3}
&=(t+2)(10 t^4 - 31 t^3 - 14 t^2 - 30 t + 113)\\
&=10(t^5+2t+2)+(-11 t^4 - 76 t^3 - 58 t^2 + 33 t + 206)\\
&=-11 t^4 - 76 t^3 - 58 t^2 + 33 t + 206
\end{align}
$$
