Question

Obter o quociente e o resto da divisao de f(x)=x elevado a 5
-3x³ +2x² +4 por g (x)=x+1

Answer 1

$f(x)=x^{5}-3x^{3}+2x^{2}+4\\ \\ =\underbrace{x^5+x^4}-x^4-3x^{3}+2x^{2}+4\\ \\ =x^{4}\left(x+1 \right )-x^{4}-3x^{3}+2x^{2}+4\\ \\ =x^{4}\left(x+1 \right )\underbrace{-x^{4}-x^{3}}+x^{3}-3x^{3}+2x^{2}+4\\ \\ =x^{4}\left(x+1 \right )-x^{3}\left(x+1 \right )-2x^{3}+2x^{2}+4\\ \\ =x^{4}\left(x+1 \right )-x^{3}\left(x+1 \right )\underbrace{-2x^{3}-2x^{2}}+2x^{2}+2x^{2}+4\\ \\ =x^{4}\left(x+1 \right )-x^{3}\left(x+1 \right )-2x^{2}\left(x+1 \right )+4x^{2}+4\\ \\ =x^{4}\left(x+1 \right )-x^{3}\left(x+1 \right )-2x^{2}\left(x+1 \right )\underbrace{+4x^{2}+4x}-4x+4\\ \\ =x^{4}\left(x+1 \right )-x^{3}\left(x+1 \right )-2x^{2}\left(x+1 \right )+4x\left(x+1 \right )-4x+4\\ \\ =x^{4}\left(x+1 \right )-x^{3}\left(x+1 \right )-2x^{2}\left(x+1 \right )+4x\left(x+1 \right )\underbrace{-4x-4}+4+4\\ \\ =x^{4}\left(x+1 \right )-x^{3}\left(x+1 \right )-2x^{2}\left(x+1 \right )+4x\left(x+1 \right )-4\left(x+1 \right )+8$

$=\left(x+1 \right )\underbrace{\left(x^{4}-x^{3}-2x^{2}+4x-4 \right )}_{\text{quociente}}+\underbrace{8}_{\text{resto}}$