Question

On dividing $f\left(x\right)$ by a polynomial $x-1-x^2$, the quotient $q\left(x\right)$ and remainder $r\left(x\right)$ are $(x-2)$ and $3$ respectively. Then $f\left(x\right)$ is

• A. $f\left(x\right)=-3x^2-x+7$
• B. $f\left(x\right)=-x^3+x^2-x+7$
• C. $f\left(x\right)=3x^2-3x+5$
• D. $f\left(x\right)=-x^3+3x^2-3x+5$

Answer 1

The correct option is D $f\left(x\right)=-x^3+3x^2-3x+5$

$f\left(x\right)=q\left(x\right)g\left(x\right)+r\left(x\right)$
$\therefore f\left(x\right)=(x-2)(x-1-x^2)+3$
$\Rightarrow f\left(x\right)=x(x-1-x^2)-2(x-1-x^2)+3$
$=x^2-x-x^3-2x+2+2x^2+3$
$=-x^3+3x^2-3x+5$