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

Queremos saber o quociente $Q(x)$ e o resto $R(x)$ da divisão do polinômio $P(x)$ por $g(x)=x+1$, onde $P(x)=x^{5}-3x^{3}+2x^{2}+4$.

Sendo assim, queremos escrever

$P(x)=g(x) \cdot Q(x)+R(x)$


$P(x)=x^{5}-3x^{3}+2x^{2}+4\\ \\ =x^{5}+x^{4}-x^{4}-3x^{3}+2x^{2}+4\\ \\ =x^{4}(x+1)-x^{4}-3x^{3}+2x^{2}+4\\ \\ =x^{4}(x+1)-x^{4}-x^{3}-2x^{3}+2x^{2}+4\\ \\ =x^{4}(x+1)-x^{3}(x+1)-2x^{3}+2x^{2}+4\\ \\ =x^{4}(x+1)-x^{3}(x+1)-2x^{3}-2x^{2}+4x^{2}+4\\ \\ =x^{4}(x+1)-x^{3}(x+1)-2x^{2}(x+1)+4x^{2}+4\\ \\ =x^{4}(x+1)-x^{3}(x+1)-2x^{2}(x+1)+4x^{2}+4x-4x+4\\ \\ =x^{4}(x+1)-x^{3}(x+1)-2x^{2}(x+1)+4x(x+1)-4x+4\\ \\ =x^{4}(x+1)-x^{3}(x+1)-2x^{2}(x+1)+4x(x+1)-4x-4+8\\ \\ =x^{4}(x+1)-x^{3}(x+1)-2x^{2}(x+1)+4x(x+1)-4(x+1)+8\\ \\ =(x+1)\underbrace{\left(x^{4}-x^{3}-2x^{2}+4x-4)}_{Q(x)}+\underbrace{8}_{R(x)}$

Logo

$Q(x)=x^{4}-x^{3}-2x^{2}+4x-4 \text{ (quociente)}\\ \\ R(x)=8 \text{ (resto)}$