Divida $$$2 x^{4} - 3 x^{3} - 15 x^{2} + 32 x - 12$$$ por $$$x^{2} - 4 x - 12$$$

Escreva o problema no formato especial:

$$$\begin{array}{r|r}\hline\\x^{2}- 4 x-12&2 x^{4}- 3 x^{3}- 15 x^{2}+32 x-12\end{array}$$$

Passo 1

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

Anote o resultado calculado na parte superior da tabela.

Multiplique pelo divisor: $$$2 x^{2} \left(x^{2}- 4 x-12\right) = 2 x^{4}- 8 x^{3}- 24 x^{2}$$$.

Subtraia o dividendo do resultado obtido: $$$\left(2 x^{4}- 3 x^{3}- 15 x^{2}+32 x-12\right) - \left(2 x^{4}- 8 x^{3}- 24 x^{2}\right) = 5 x^{3}+9 x^{2}+32 x-12.$$$

$$\begin{array}{r|rrrrr:c}&{\color{Green}2 x^{2}}&&&&&\\\hline\\{\color{Magenta}x^{2}}- 4 x-12&{\color{Green}2 x^{4}}&- 3 x^{3}&- 15 x^{2}&+32 x&-12&\frac{{\color{Green}2 x^{4}}}{{\color{Magenta}x^{2}}} = {\color{Green}2 x^{2}}\\&-\phantom{2 x^{4}}&&&&&\\&2 x^{4}&- 8 x^{3}&- 24 x^{2}&&&{\color{Green}2 x^{2}} \left(x^{2}- 4 x-12\right) = 2 x^{4}- 8 x^{3}- 24 x^{2}\\\hline\\&&5 x^{3}&+9 x^{2}&+32 x&-12&\end{array}$$

Passo 2

Divida o termo principal do resto obtido pelo termo principal do divisor: $$$\frac{5 x^{3}}{x^{2}} = 5 x$$$

Anote o resultado calculado na parte superior da tabela.

Multiplique pelo divisor: $$$5 x \left(x^{2}- 4 x-12\right) = 5 x^{3}- 20 x^{2}- 60 x$$$.

Subtraia o resto do resultado obtido: $$$\left(5 x^{3}+9 x^{2}+32 x-12\right) - \left(5 x^{3}- 20 x^{2}- 60 x\right) = 29 x^{2}+92 x-12$$$.

$$\begin{array}{r|rrrrr:c}&2 x^{2}&{\color{BlueViolet}+5 x}&&&&\\\hline\\{\color{Magenta}x^{2}}- 4 x-12&2 x^{4}&- 3 x^{3}&- 15 x^{2}&+32 x&-12&\\&-\phantom{2 x^{4}}&&&&&\\&2 x^{4}&- 8 x^{3}&- 24 x^{2}&&&\\\hline\\&&{\color{BlueViolet}5 x^{3}}&+9 x^{2}&+32 x&-12&\frac{{\color{BlueViolet}5 x^{3}}}{{\color{Magenta}x^{2}}} = {\color{BlueViolet}5 x}\\&&-\phantom{5 x^{3}}&&&&\\&&5 x^{3}&- 20 x^{2}&- 60 x&&{\color{BlueViolet}5 x} \left(x^{2}- 4 x-12\right) = 5 x^{3}- 20 x^{2}- 60 x\\\hline\\&&&29 x^{2}&+92 x&-12&\end{array}$$

Passo 3

Divida o termo principal do resto obtido pelo termo principal do divisor: $$$\frac{29 x^{2}}{x^{2}} = 29$$$

Anote o resultado calculado na parte superior da tabela.

Multiplique pelo divisor: $$$29 \left(x^{2}- 4 x-12\right) = 29 x^{2}- 116 x-348$$$.

Subtraia o resto do resultado obtido: $$$\left(29 x^{2}+92 x-12\right) - \left(29 x^{2}- 116 x-348\right) = 208 x+336$$$.

$$\begin{array}{r|rrrrr:c}&2 x^{2}&+5 x&{\color{Brown}+29}&&&\\\hline\\{\color{Magenta}x^{2}}- 4 x-12&2 x^{4}&- 3 x^{3}&- 15 x^{2}&+32 x&-12&\\&-\phantom{2 x^{4}}&&&&&\\&2 x^{4}&- 8 x^{3}&- 24 x^{2}&&&\\\hline\\&&5 x^{3}&+9 x^{2}&+32 x&-12&\\&&-\phantom{5 x^{3}}&&&&\\&&5 x^{3}&- 20 x^{2}&- 60 x&&\\\hline\\&&&{\color{Brown}29 x^{2}}&+92 x&-12&\frac{{\color{Brown}29 x^{2}}}{{\color{Magenta}x^{2}}} = {\color{Brown}29}\\&&&-\phantom{29 x^{2}}&&&\\&&&29 x^{2}&- 116 x&-348&{\color{Brown}29} \left(x^{2}- 4 x-12\right) = 29 x^{2}- 116 x-348\\\hline\\&&&&208 x&+336&\end{array}$$

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

A tabela resultante é mostrada mais uma vez:

$$\begin{array}{r|rrrrr:c}&{\color{Green}2 x^{2}}&{\color{BlueViolet}+5 x}&{\color{Brown}+29}&&&\text{Dicas}\\\hline\\{\color{Magenta}x^{2}}- 4 x-12&{\color{Green}2 x^{4}}&- 3 x^{3}&- 15 x^{2}&+32 x&-12&\frac{{\color{Green}2 x^{4}}}{{\color{Magenta}x^{2}}} = {\color{Green}2 x^{2}}\\&-\phantom{2 x^{4}}&&&&&\\&2 x^{4}&- 8 x^{3}&- 24 x^{2}&&&{\color{Green}2 x^{2}} \left(x^{2}- 4 x-12\right) = 2 x^{4}- 8 x^{3}- 24 x^{2}\\\hline\\&&{\color{BlueViolet}5 x^{3}}&+9 x^{2}&+32 x&-12&\frac{{\color{BlueViolet}5 x^{3}}}{{\color{Magenta}x^{2}}} = {\color{BlueViolet}5 x}\\&&-\phantom{5 x^{3}}&&&&\\&&5 x^{3}&- 20 x^{2}&- 60 x&&{\color{BlueViolet}5 x} \left(x^{2}- 4 x-12\right) = 5 x^{3}- 20 x^{2}- 60 x\\\hline\\&&&{\color{Brown}29 x^{2}}&+92 x&-12&\frac{{\color{Brown}29 x^{2}}}{{\color{Magenta}x^{2}}} = {\color{Brown}29}\\&&&-\phantom{29 x^{2}}&&&\\&&&29 x^{2}&- 116 x&-348&{\color{Brown}29} \left(x^{2}- 4 x-12\right) = 29 x^{2}- 116 x-348\\\hline\\&&&&208 x&+336&\end{array}$$

Portanto, $$$\frac{2 x^{4} - 3 x^{3} - 15 x^{2} + 32 x - 12}{x^{2} - 4 x - 12} = \left(2 x^{2} + 5 x + 29\right) + \frac{208 x + 336}{x^{2} - 4 x - 12}.$$$