Question

seja Q(x) o quociente da divisão do polinômio P(x)=x^4-1 pelo polinômio D(x)=x-1 , é correto afirmar.

Answer 1

$x^{4}$ -1              | x-1
$-x^{4}$ +x³ -1     x³+x²+x+1
        -x³-x²-1
             x²+x-1  
                 -x+1
                     0

Q(x) = x³-x²+x+1

Answer 2

$P(x)=x^4-1\to~~~x^4+0x^3+0x^2+x-1\\\\ D(x)=x-1 \\\\\\P(x):D(x)=Q(x)$



$~~~x^4+0x^3+0x^2+0x-1~|\underline{~x-1~~~~}\\ \underline{-x^4~+~x^3~~~~~~~~~~~~~~~~~~~~} ~~~~{ \it x^3+x^2+x+1}\\~~~~~~~~~~~~x^3+0x^2+0x-1\\ ~~~~~~~~~~ \underline{-x^3+~x^2~~~~~~~~~~~~}\\~~~~~~~~~~~~~~~~~~~~~x^2+0x-1 \\~~~~~~~~~~~~~~~~~~ \underline{-x^2~+~x~~~~~} \\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ x-1\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~ \underline{-x+1} \\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0$


$\large\boxed{{ \it Q(x) = x^3+x^2+x+1} }$