Expressiong $\frac{t+2}{t^3+3}$ in the form $a_o+a_1t+...+a_4t^4$, where $t$ is a root of $x^5+2x+2$

Question

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!

Answer 1

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, $$ \begin{align} 367\frac{t+2}{t^3+3} &=(t+2)(10 t^4 - 31 t^3 - 14 t^2 - 30 t + 113)\\ &=10(t^5+2t+2)+(-11 t^4 - 76 t^3 - 58 t^2 + 33 t + 206)\\ &=-11 t^4 - 76 t^3 - 58 t^2 + 33 t + 206 \end{align} $$

Answer 2

$$ \left( x^{3} + 3 \right) \left( \frac{ 10 x^{4} - 31 x^{3} - 14 x^{2} - 30 x + 113 }{ 367 } \right) \equiv 1 \pmod { x^{5} + 2 x + 2 } $$

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$$ \left( x^{5} + 2 x + 2 \right) $$

$$ \left( x^{3} + 3 \right) $$

$$ \left( x^{5} + 2 x + 2 \right) = \left( x^{3} + 3 \right) \cdot \color{magenta}{ \left( x^{2} \right) } + \left( - 3 x^{2} + 2 x + 2 \right) $$ $$ \left( x^{3} + 3 \right) = \left( - 3 x^{2} + 2 x + 2 \right) \cdot \color{magenta}{ \left( \frac{ - 3 x - 2 }{ 9 } \right) } + \left( \frac{ 10 x + 31 }{ 9 } \right) $$ $$ \left( - 3 x^{2} + 2 x + 2 \right) = \left( \frac{ 10 x + 31 }{ 9 } \right) \cdot \color{magenta}{ \left( \frac{ - 270 x + 1017 }{ 100 } \right) } + \left( \frac{ -3303}{100 } \right) $$ $$ \left( \frac{ 10 x + 31 }{ 9 } \right) = \left( \frac{ -3303}{100 } \right) \cdot \color{magenta}{ \left( \frac{ - 1000 x - 3100 }{ 29727 } \right) } + \left( 0 \right) $$ $$ \frac{ 0}{1} $$ $$ \frac{ 1}{0} $$ $$ \color{magenta}{ \left( x^{2} \right) } \Longrightarrow \Longrightarrow \frac{ \left( x^{2} \right) }{ \left( 1 \right) } $$ $$ \color{magenta}{ \left( \frac{ - 3 x - 2 }{ 9 } \right) } \Longrightarrow \Longrightarrow \frac{ \left( \frac{ - 3 x^{3} - 2 x^{2} + 9 }{ 9 } \right) }{ \left( \frac{ - 3 x - 2 }{ 9 } \right) } $$ $$ \color{magenta}{ \left( \frac{ - 270 x + 1017 }{ 100 } \right) } \Longrightarrow \Longrightarrow \frac{ \left( \frac{ 90 x^{4} - 279 x^{3} - 126 x^{2} - 270 x + 1017 }{ 100 } \right) }{ \left( \frac{ 90 x^{2} - 279 x - 126 }{ 100 } \right) } $$ $$ \color{magenta}{ \left( \frac{ - 1000 x - 3100 }{ 29727 } \right) } \Longrightarrow \Longrightarrow \frac{ \left( \frac{ - 100 x^{5} - 200 x - 200 }{ 3303 } \right) }{ \left( \frac{ - 100 x^{3} - 300 }{ 3303 } \right) } $$ $$ \left( x^{5} + 2 x + 2 \right) \left( \frac{ 10 x^{2} - 31 x - 14 }{ 367 } \right) - \left( x^{3} + 3 \right) \left( \frac{ 10 x^{4} - 31 x^{3} - 14 x^{2} - 30 x + 113 }{ 367 } \right) = \left( -1 \right) $$