Divida $$$x^{6}$$$ por $$$\left(x^{2} + 1\right)^{2}$$$

Reescreva o divisor: $$$\left(x^{2} + 1\right)^{2} = x^{4} + 2 x^{2} + 1$$$.

Escreva o problema no formato especial (os termos ausentes são escritos com coeficientes nulos):

$$$\begin{array}{r|r}\hline\\x^{4}+2 x^{2}+1&x^{6}+0 x^{5}+0 x^{4}+0 x^{3}+0 x^{2}+0 x+0\end{array}$$$

Passo 1

Divida o termo principal do dividendo pelo termo principal do divisor: $$$\frac{x^{6}}{x^{4}} = x^{2}$$$.

Escreva o resultado calculado na parte superior da tabela.

Multiplique-o pelo divisor: $$$x^{2} \left(x^{4}+2 x^{2}+1\right) = x^{6}+2 x^{4}+x^{2}$$$.

Subtraia o dividendo do resultado obtido: $$$\left(x^{6}\right) - \left(x^{6}+2 x^{4}+x^{2}\right) = - 2 x^{4}- x^{2}$$$.

$$\begin{array}{r|rrrrrrr:c}&{\color{DarkBlue}x^{2}}&&&&&&&\\\hline\\{\color{Magenta}x^{4}}+2 x^{2}+1&{\color{DarkBlue}x^{6}}&+0 x^{5}&+0 x^{4}&+0 x^{3}&+0 x^{2}&+0 x&+0&\frac{{\color{DarkBlue}x^{6}}}{{\color{Magenta}x^{4}}} = {\color{DarkBlue}x^{2}}\\&-\phantom{x^{6}}&&&&&&&\\&x^{6}&+0 x^{5}&+2 x^{4}&+0 x^{3}&+x^{2}&&&{\color{DarkBlue}x^{2}} \left(x^{4}+2 x^{2}+1\right) = x^{6}+2 x^{4}+x^{2}\\\hline\\&&&- 2 x^{4}&+0 x^{3}&- x^{2}&+0 x&+0&\end{array}$$

Passo 2

Divida o termo de maior grau do resto obtido pelo termo de maior grau do divisor: $$$\frac{- 2 x^{4}}{x^{4}} = -2$$$.

Escreva o resultado calculado na parte superior da tabela.

Multiplique-o pelo divisor: $$$- 2 \left(x^{4}+2 x^{2}+1\right) = - 2 x^{4}- 4 x^{2}-2$$$.

Subtraia o resto do resultado obtido: $$$\left(- 2 x^{4}- x^{2}\right) - \left(- 2 x^{4}- 4 x^{2}-2\right) = 3 x^{2}+2$$$.

$$\begin{array}{r|rrrrrrr:c}&x^{2}&{\color{Chocolate}-2}&&&&&&\\\hline\\{\color{Magenta}x^{4}}+2 x^{2}+1&x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&+0 x^{2}&+0 x&+0&\\&-\phantom{x^{6}}&&&&&&&\\&x^{6}&+0 x^{5}&+2 x^{4}&+0 x^{3}&+x^{2}&&&\\\hline\\&&&{\color{Chocolate}- 2 x^{4}}&+0 x^{3}&- x^{2}&+0 x&+0&\frac{{\color{Chocolate}- 2 x^{4}}}{{\color{Magenta}x^{4}}} = {\color{Chocolate}-2}\\&&&-\phantom{- 2 x^{4}}&&&&&\\&&&- 2 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&-2&{\color{Chocolate}-2} \left(x^{4}+2 x^{2}+1\right) = - 2 x^{4}- 4 x^{2}-2\\\hline\\&&&&&3 x^{2}&+0 x&+2&\end{array}$$

Como o grau do resto é menor que o grau do divisor, terminamos.

A tabela resultante é mostrada novamente:

$$\begin{array}{r|rrrrrrr:c}&{\color{DarkBlue}x^{2}}&{\color{Chocolate}-2}&&&&&&\text{Dicas}\\\hline\\{\color{Magenta}x^{4}}+2 x^{2}+1&{\color{DarkBlue}x^{6}}&+0 x^{5}&+0 x^{4}&+0 x^{3}&+0 x^{2}&+0 x&+0&\frac{{\color{DarkBlue}x^{6}}}{{\color{Magenta}x^{4}}} = {\color{DarkBlue}x^{2}}\\&-\phantom{x^{6}}&&&&&&&\\&x^{6}&+0 x^{5}&+2 x^{4}&+0 x^{3}&+x^{2}&&&{\color{DarkBlue}x^{2}} \left(x^{4}+2 x^{2}+1\right) = x^{6}+2 x^{4}+x^{2}\\\hline\\&&&{\color{Chocolate}- 2 x^{4}}&+0 x^{3}&- x^{2}&+0 x&+0&\frac{{\color{Chocolate}- 2 x^{4}}}{{\color{Magenta}x^{4}}} = {\color{Chocolate}-2}\\&&&-\phantom{- 2 x^{4}}&&&&&\\&&&- 2 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&-2&{\color{Chocolate}-2} \left(x^{4}+2 x^{2}+1\right) = - 2 x^{4}- 4 x^{2}-2\\\hline\\&&&&&3 x^{2}&+0 x&+2&\end{array}$$

Portanto, $$$\frac{x^{6}}{\left(x^{2} + 1\right)^{2}} = \left(x^{2} - 2\right) + \frac{3 x^{2} + 2}{\left(x^{2} + 1\right)^{2}}$$$.