Question

If the polynomial $P\left(x\right)=x^3-3x^2+x+2$ is divided by $5\left(x\right)$ then the quatient polynomial $x-2$ and the reminder polynomial $-2x+4$ are obtained. Find the polynomial $5\left(x\right).$

Answer 1

$(x^3-3x^2+x+2)=(x-2)S\left(x\right)+(-2x+4)$

$x^3-3x^2+x+2+2x-4=(x-2)S\left(x\right)$

$x^3-3x^2+3x-2=(x-2)S\left(x\right)$

$(x-2)(x^2-x+1)=(x-2)S\left(x\right)$

$S\left(x\right)=x^2-x+1$